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Sectional curvature

In Riemannian geometry, the sectional curvature is one of the ways to describe the curvature of Riemannian manifolds. The sectional curvature K(σp) depends on a two-dimensional linear subspace σp of the tangent space at a point p of the manifold. It can be defined geometrically as the Gaussian curvature of the surface which has the plane σp as a tangent plane at p, obtained from geodesics which start at p in the directions of σp (in other words, the image of σp under the exponential map at p). The sectional curvature is a real-valued function on the 2-Grassmannian bundle over the manifold.

The sectional curvature determines the curvature tensor completely.

Definition
Given a Riemannian manifold and two linearly independent tangent vectors at the same point, u and v, we can define

 * $$K(u,v)={\langle R(u,v)v,u\rangle\over \langle u,u\rangle\langle v,v\rangle-\langle u,v\rangle^2}$$

Here R is the Riemann curvature tensor, defined here by the convention $$R(u,v)w=\nabla_u\nabla_vw-\nabla_v\nabla_uw-\nabla_{[u,v]}w.$$ Some sources use the opposite convention $$R(u,v)w=\nabla_v\nabla_uw-\nabla_u\nabla_vw-\nabla_{[v,u]}w,$$ in which case K(u,v) must be defined with $$\langle R(u,v)u,v\rangle$$ in the numerator instead of $$\langle R(u,v)v,u\rangle.$$

Note that the linear independence of u and v forces the denominator in the above expression to be nonzero, so that K(u,v) is well-defined. In particular, if u and v are orthonormal, then the definition takes on the simple form
 * $$K(u,v) = \langle R(u,v)v,u\rangle.$$

It is straightforward to check that if $$u,v\in T_pM$$ are linearly independent and span the same two-dimensional linear subspace of the tangent space $$T_pM$$ as $$x,y\in T_pM$$, then $$K(u,v)=K(x,y).$$ So one may consider the sectional curvature as a real-valued function whose input is a two-dimensional linear subspace of a tangent space.

Alternative definitions
Alternatively, the sectional curvature can be characterized by the circumference of small circles. Let $$P$$ be a two-dimensional plane in $$T_xM$$. Let $$C_P(r)$$ for sufficiently small $$r > 0$$ denote the image under the exponential map at $$p$$ of the unit circle in $$P$$, and let $$l_P(r)$$ denote the length of $$C_P(r)$$. Then it can be proven that

 * $$l_P(r)=2\pi r \left(1-{r^2\over 6}\sigma(P)+O(r^3)\right),$$

as $$r \to 0$$, for some number $$\sigma(P)$$. This number $$\sigma(P)$$ at $$p$$ is the sectional curvature of $$P$$ at $$p$$.

Manifolds with constant sectional curvature
One says that a Riemannian manifold has "constant curvature $$\kappa$$" if $$\operatorname{sec}(P)=\kappa$$ for all two-dimensional linear subspaces $$P\subset T_pM$$ and for all $$p\in M.$$

The Schur lemma states that if (M,g) is a connected Riemannian manifold with dimension at least three, and if there is a function $$f:M\to\mathbb{R}$$ such that $$\operatorname{sec}(P)=f(p)$$ for all two-dimensional linear subspaces $$P\subset T_pM$$ and for all $$p\in M,$$ then f must be constant and hence (M,g) has constant curvature.

A Riemannian manifold with constant sectional curvature is called a space form. If $$\kappa$$ denotes the constant value of the sectional curvature, then the curvature tensor can be written as
 * $$R(u,v)w=\kappa \big(\langle v,w\rangle u-\langle u,w\rangle v\big)$$

for any $$u,v,w\in T_pM.$$

Since any Riemannian metric is parallel with respect to its Levi-Civita connection, this shows that the Riemann tensor of any constant-curvature space is also parallel. The Ricci tensor is then given by $$\operatorname{Ric} = (n - 1)\kappa g$$ and the scalar curvature is $$n(n - 1)\kappa.$$ In particular, any constant-curvature space is Einstein and has constant scalar curvature.

The model examples
Given a positive number $$a,$$ define In the usual terminology, these Riemannian manifolds are referred to as Euclidean space, the n-sphere, and hyperbolic space. Here, the point is that each is a complete connected smooth Riemannian manifold with constant curvature. To be precise, the Riemannian metric $$g_{\mathbb{R}^n}$$ has constant curvature 0, the Riemannian metric $$g_{S^n(a)}$$ has constant curvature $$1/a^2,$$ and the Riemannian metric $$g_{H^n(a)}$$ has constant curvature $$-1/a^2.$$
 * $$\left(\mathbb{R}^n, g_{\mathbb{R}^n}\right)$$ to be the standard Riemannian structure
 * $$\left(S^n(a), g_{S^n(a)}\right)$$ to be the sphere $$S^n(a) \equiv \left\{x\in\mathbb{R}^{n+1}: |x| = a\right\}$$ with $$g_{S^n(a)}$$ given by the pullback of the standard Riemannian structure on $$\mathbb{R}^{n+1}$$ by the inclusion map $$S^n(a) \to \mathbb{R}^{n+1}$$
 * $$\left(H^n(a), g_{H^n(a)}\right)$$ to be the ball $$H^n(a) \equiv \left\{x\in\mathbb{R}^n: |x| < a\right\}$$ with $$g_{H^n(a)} = a^2\frac{\bigl(a^2 - |x|{}^2\bigr)\left(dx_1^2 + \cdots + dx_n^2\right) - \left(x_1 dx_1 + \cdots + x_n dx_n\right)^2}{\bigl(a^2 - |x|{}^2\bigr)^2}.$$

Furthermore, these are the 'universal' examples in the sense that if $$(M, g)$$ is a smooth, connected, and simply-connected complete Riemannian manifold with constant curvature, then it is isometric to one of the above examples; the particular example is dictated by the value of the constant curvature of $$g,$$ according to the constant curvatures of the above examples.

If $$(M, g)$$ is a smooth and connected complete Riemannian manifold with constant curvature, but is not assumed to be simply-connected, then consider the universal covering space $$\pi:\widetilde{M}\to M$$ with the pullback Riemannian metric $$\pi^\ast g.$$ Since $$\pi$$ is, by topological principles, a covering map, the Riemannian manifold $$(\widetilde{M},\pi^\ast g)$$ is locally isometric to $$(M,g)$$, and so it is a smooth, connected, and simply-connected complete Riemannian manifold with the same constant curvature as $$g.$$ It must then be isometric one of the above model examples. Note that the deck transformations of the universal cover are isometries relative to the metric $$\pi^\ast g.$$

The study of Riemannian manifolds with constant negative curvature is called hyperbolic geometry.

Scaling
Let $$(M, g)$$ be a smooth manifold, and let $$\lambda$$ be a positive number. Consider the Riemannian manifold $$(M, \lambda g).$$ The curvature tensor, as a multilinear map $$T_pM\times T_pM\times T_pM\to T_pM,$$ is unchanged by this modification. Let $$v,w$$ be linearly independent vectors in $$T_pM$$. Then
 * $$K_{\lambda g}(v, w) = \frac{\lambda g\left(R^{\lambda g}(v, w)w, v\right)}{|v|_{\lambda g}^2|w|_{\lambda g}^2 - \langle v, w \rangle\vphantom|_{\lambda g}^2} = \frac{1}{\lambda}\frac{g\left(R^g(v, w)w, v\right)}{|v|_g^2|w|_g^2 - \langle v, w \rangle\vphantom|_g^2} = \frac{1}{\lambda}K_g(v,w).$$

So multiplication of the metric by $$\lambda$$ multiplies all of the sectional curvatures by $$\lambda^{-1}.$$

Toponogov's theorem
Toponogov's theorem affords a characterization of sectional curvature in terms of how "fat" geodesic triangles appear when compared to their Euclidean counterparts. The basic intuition is that, if a space is positively curved, then the edge of a triangle opposite some given vertex will tend to bend away from that vertex, whereas if a space is negatively curved, then the opposite edge of the triangle will tend to bend towards the vertex.

More precisely, let M be a complete Riemannian manifold, and let xyz be a geodesic triangle in M (a triangle each of whose sides is a length-minimizing geodesic). Finally, let m be the midpoint of the geodesic xy. If M has non-negative curvature, then for all sufficiently small triangles
 * $$d(z,m)^2 \ge \tfrac12 d(z,x)^2 + \tfrac12 d(z,y)^2 - \tfrac14 d(x,y)^2$$

where d is the distance function on M. The case of equality holds precisely when the curvature of M vanishes, and the right-hand side represents the distance from a vertex to the opposite side of a geodesic triangle in Euclidean space having the same side-lengths as the triangle xyz. This makes precise the sense in which triangles are "fatter" in positively curved spaces. In non-positively curved spaces, the inequality goes the other way:
 * $$d(z,m)^2 \le \tfrac12 d(z,x)^2 + \tfrac12 d(z,y)^2 - \tfrac14 d(x,y)^2.$$

If tighter bounds on the sectional curvature are known, then this property generalizes to give a comparison theorem between geodesic triangles in M and those in a suitable simply connected space form; see Toponogov's theorem. Simple consequences of the version stated here are:

 * A complete Riemannian manifold has non-negative sectional curvature if and only if the function $$f_p(x) = \operatorname{dist}^2(p,x)$$ is 1-concave for all points p.
 * A complete simply connected Riemannian manifold has non-positive sectional curvature if and only if the function $$f_p(x) = \operatorname{dist}^2(p,x)$$ is 1-convex.

Manifolds with non-positive sectional curvature
In 1928, Élie Cartan proved the Cartan–Hadamard theorem: if M is a complete manifold with non-positive sectional curvature, then its universal cover is diffeomorphic to a Euclidean space. In particular, it is aspherical: the homotopy groups $$\pi_i(M)$$ for i ≥ 2 are trivial. Therefore, the topological structure of a complete non-positively curved manifold is determined by its fundamental group. Preissman's theorem restricts the fundamental group of negatively curved compact manifolds. The Cartan–Hadamard conjecture states that the classical isoperimetric inequality should hold in all simply connected spaces of non-positive curvature, which are called Cartan-Hadamard manifolds.

Manifolds with positive sectional curvature
Little is known about the structure of positively curved manifolds. The soul theorem implies that a complete non-compact non-negatively curved manifold is diffeomorphic to a normal bundle over a compact non-negatively curved manifold. As for compact positively curved manifolds, there are two classical results:

 * It follows from the Myers theorem that the fundamental group of such a manifold is finite.
 * It follows from the Synge theorem that the fundamental group of such a manifold in even dimensions is 0, if orientable and $$\mathbb Z_2$$ otherwise. In odd dimensions a positively curved manifold is always orientable.

Moreover, there are relatively few examples of compact positively curved manifolds, leaving a lot of conjectures (e.g., the Hopf conjecture on whether there is a metric of positive sectional curvature on $$\mathbb S^2 \times \mathbb S^2$$). The most typical way of constructing new examples is the following corollary from the O'Neill curvature formulas: if $$(M, g)$$ is a Riemannian manifold admitting a free isometric action of a Lie group G, and M has positive sectional curvature on all 2-planes orthogonal to the orbits of G, then the manifold $$M/G$$ with the quotient metric has positive sectional curvature. This fact allows one to construct the classical positively curved spaces, being spheres and projective spaces, as well as these examples :

 * The Berger spaces $$B^7=SO(5)/SO(3)$$ and $$B^{13}=SU(5)/\operatorname{Sp}(2) \cdot \mathbb S^1$$.
 * The Wallach spaces (or the homogeneous flag manifolds): $$W^6=SU(3)/T^2$$, $$W^{12}=\operatorname{Sp}(3)/\operatorname{Sp}(1)^3$$ and $$W^{24}=F_4/\operatorname{Spin}(8)$$.
 * The Aloff–Wallach spaces $$W^7_{p,q} = SU(3)/\operatorname{diag}\left(z^p, z^q, \overline{z}^{p+q}\right)$$.
 * The Eschenburg spaces $$E_{k,l} = \operatorname{diag}\left(z^{k_1}, z^{k_2}, z^{k_3}\right)\backslash SU(3)/\operatorname{diag}\left(z^{l_1}, z^{l_2}, z^{l_3}\right)^{-1}.$$
 * The Bazaikin spaces $$B^{13}_p = \operatorname{diag}\left(z_1^{p_1}, \dots, z_1^{p_5}\right)\backslash U(5)/\operatorname{diag}(z_2 A, 1)^{-1}$$, where $$A\in \operatorname{Sp}(2)\subset SU(4)$$.

Manifolds with non-negative sectional curvature
Cheeger and Gromoll proved their soul theorem which states that any non-negatively curved complete non-compact manifold $$M$$ has a totally convex compact submanifold $$S$$ such that $$M$$ is diffeomorphic to the normal bundle of $$S$$. Such an $$S$$ is called the soul of $$M$$. In particular, this theorem implies that $$M$$ is homotopic to its soul $$S$$ which has the dimension less than $$M$$.

title: Sectional curvature


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Talk:Vitellaria

Corret name is Vitellaria?
I found the following in the International Code of Botanical Nomenclature: this does work

Ex. 2. The holotype of Vitellaria paradoxa C. F. Gaertn. (1807) is a seed of unknown provenance (P), clearly belonging to the species currently known as Butyrospermum paradoxum (C. F. Gaertn.) Hepper. However, the two subspecies recognized within that species can only be distinguished by characters of foliage or inflorescence. Hall & Hurdle (in Taxon 44: 410. 1995) designated an epitype with foliage, Mungo Park (BM). It belongs to the western subspecies, now to be known as B. paradoxum subsp. paradoxum.

 * A rare case of an error in the ICBN. If you look up Butyrospermum you will see it is no longer the name of a current genus: Vitellaria has priority. A proposal-to-conserve failed (narrowly). Very famous case. A pity, as the world continues to use Butyrospermum parkii. Brya 19:19, 23 September 2005 (UTC)
 * Assuming this is correct, should't we move the article to Vitellaria paradoxa (or, probably better, "shea tree"?) --Jorge Stolfi (talk) 21:24, 21 December 2009 (UTC)

Then why isn't the article titled Vitellaria? Am I missing something? — Preceding unsigned comment added by 174.31.115.210 (talk) 23:20, 19 March 2012 (UTC)

 * I see the article is now titled with the right name, but while the previous name B. parkii is explained in the Etymology section, the current name isn't. What is Vitellaria named for? What was so paradoxical about it?  Did C. F. Gaertner give it both names?  When?  Was he trying to change the name he'd earlier given, or just didn't realize it was a kind he'd already named? How does Lucuma fit into the timeline?  The list of names below (under the heading "Scarcity?") is informative but I can't quite follow its format because the list is run together into a paragraph. - Egmonster (talk) 14:20, 17 July 2018 (UTC)

 * The book "100 Top Food Plants" has some information on the naming see page 474. According to that, Vitellaria derives from vitellus (yolk in latin), a reference to the valuable kernel and aria meaning 'like' and the paradoxa specific name (meaning strange or contrary to expectation) is from its ability to grow in extremely dry regions. The Hall 1996 monograph probably goes into a lot more detail. Mikenorton (talk) 08:38, 28 August 2018 (UTC)

redirect
Why does anything to do with Vitellaria redirect to Butyrospermum parkii????! This is not the priority name and has not been for some time. Most botanists recognize both but we now use Vitellaria paradoxa, even though--I'll concede--the former name is a bit more descriptive (as long as one knows Greek roots). It should be the other way around, please. Follow convention. When wiki does its own thing it borders on looking ridiculous and is part of the reason it is sometimes scorned. Butyrospermum (sp.) should redirect to Vitellaria (sp.). Thank you.
 * See above. --Jorge Stolfi (talk) 22:37, 21 December 2009 (UTC)

Spelling
"...and ghariti in the Malian language"

 * This line originally had a spelling question in it: "...and ghariti (sp?) in the Malian language". I removed it, but did not check the spelling. I am noting it here in case anyone wants to research it. --DanielCD 19:53, 12 May 2005 (UTC)

Language names
There is neither a "Senegalese language" nor a "Malian language." I've tried finding the language names on the Internet, to no avail. Someone please correct this. -- Tintazul 15:41, 30 March 2006 (UTC)
 * Hi Tintazul, there are multiple languages in these countries and I couldn't find a reference to the specific one.  I have reworded to avoid referring to the languages. - BanyanTree 15:50, 30 March 2006 (UTC)
 * This issue appears to have been fixed sometime ago. --Jorge Stolfi (talk) 22:38, 21 December 2009 (UTC)

contribution
I have removed a contribution that replaced the old content but was specific to the butter rather than the plant and for which I have POV concerns. I'll paste the content below:
 * I have removed the info as the user in question proved the invalidity of it by trying to spam it into unrelated articles. - BanyanTree 19:52, 28 April 2006 (UTC)

Vandalism
Deleted some insertions by User:DaFool. This would be a good time for some admin action. --Oop 12:06, 16 August 2006 (UTC)

WikiProject Food and drink Tagging
This article talk page was automatically added with WikiProject Food and drink banner as it falls under Category:Food or one of its subcategories. If you find this addition an error, Kindly undo the changes and update the inappropriate categories if needed. The bot was instructed to tagg these articles upon consenus from WikiProject Food and drink. You can find the related request for tagging here. Maximum and careful attention was done to avoid any wrongly tagging any categories, but mistakes may happen... If you have concerns, please inform on the project talk page -- TinucherianBot (talk) 02:02, 4 July 2008 (UTC)

Redirect listed for discussion
Shea, which currently redirects to this page, has been listed at Redirects for Discussion. Your comments are welcome. Powers T 13:56, 11 October 2009 (UTC)
 * Apears to have een fixed (shea is now a disamb page). --Jorge Stolfi (talk) 22:40, 21 December 2009 (UTC)

Lucuma paradoxa var. nilotica
The following statement was in the shea butter article
 * "Many vernacular names are used for butter seed, extract - Vitellaria or Lucuma paradoxa including nilotica. "

It seems to be saying that Lucuma paradoxa var. nilotica was nother binomial name used for the plant. Is this correct? --Jorge Stolfi (talk) 21:31, 21 December 2009 (UTC)

Scarcity?
[ This item was copied from Talk:shea butter --Jorge Stolfi (talk) 22:42, 21 December 2009 (UTC) ]

Scarcity? My understanding is that shea nut trees are very common and numerous and high-yielding throughout West Africa. Any citations for the scarcity claim? 70.156.31.116 (talk) 20:33, 2 January 2008 (UTC)James D.

This article is unfortunately absolutely incomplete. There are two subspecies. Vitellaria paradoxa paradoxa and Vitellaria paradoxa nilotica

species: Vitellaria paradoxa (C.F. Gaertn.) subspecies: Vitellaria paradoxa paradoxa: subspecies: Vitellaria paradoxa nilotica (Kotschy): Synonyms: Vitellaria paradoxa niloticum Old names for Vitellaria paradoxa (C. F. Gaertn.): Butyrospermum paradoxum (C. F. Geartn.): Synonyms: Lucuma paradoxa (C.F. Gaertn.) Butyrospermum paradoxum parkii (G. Don) Synonyms: Butyrospermum parkii (G. Don) Bassia Parkii (G. Don) Bassia paradoxum Butyrospermum paradoxum niloticum (Kotschy): Synonyms: Butyrospermum niloticum (Kotschy) Butyrospermum parkii niloticum (Kotschy) Paradoxum niloticum Edited by Valentin Kurz —Preceding unsigned comment added by 92.224.48.206 (talk) 08:18, 23 August 2010 (UTC)

In Turkey too ?
Hello. There is a map showing it grows in Turkey : what about that ? T.y. Arapaima (talk) 06:33, 5 May 2011 (UTC)

Not a Sentence
"A reflection of its extensive range of occurrence nearly 5,000 km from Senegal (west) to Uganda (east) across the African continent."

I have no idea what this is supposed to mean, but it is not a sentence, as there is no verb. GeneCallahan (talk) 05:29, 1 February 2012 (UTC)

Lots of that on Wikipedia. I don't know why people try to contribute to articles when they cant even write a sentence. — Preceding unsigned comment added by 174.31.115.210 (talk) 23:24, 19 March 2012 (UTC)

Uses
Re the currently rather short section on Uses, I observe that in the UK (and presumably elsewhere) rolls of toilet paper "enriched with extracts of shea butter" are a commonplace and relatively inexpensive commodity. I leave it to others to find a suitable Reliable source. {The poster formerly known as 87.81.230.195} 90.208.127.181 (talk) 05:47, 28 August 2018 (UTC)

license: Creative Commons - Attribution Share-Alike - https://creativecommons.org/licenses/by-sa/4.0/

title: Talk:Vitellaria


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Scalar curvature

In the mathematical field of Riemannian geometry, the scalar curvature (or the Ricci scalar) is a measure of the curvature of a Riemannian manifold. To each point on a Riemannian manifold, it assigns a single real number determined by the geometry of the metric near that point. It is defined by a complicated explicit formula in terms of partial derivatives of the metric components, although it is also characterized by the volume of infinitesimally small geodesic balls. In the context of the differential geometry of surfaces, the scalar curvature is twice the Gaussian curvature, and completely characterizes the curvature of a surface. In higher dimensions, however, the scalar curvature only represents one particular part of the Riemann curvature tensor.

The definition of scalar curvature via partial derivatives is also valid in the more general setting of pseudo-Riemannian manifolds. This is significant in general relativity, where scalar curvature of a Lorentzian metric is one of the key terms in the Einstein field equations. Furthermore, this scalar curvature is the Lagrangian density for the Einstein–Hilbert action, the Euler–Lagrange equations of which are the Einstein field equations in vacuum.

The geometry of Riemannian metrics with positive scalar curvature has been widely studied. On noncompact spaces, this is the context of the positive mass theorem proved by Richard Schoen and Shing-Tung Yau in the 1970s, and reproved soon after by Edward Witten with different techniques. Schoen and Yau, and independently Mikhael Gromov and Blaine Lawson, developed a number of fundamental results on the topology of closed manifolds supporting metrics of positive scalar curvature. In combination with their results, Grigori Perelman's construction of Ricci flow with surgery in 2003 provided a complete characterization of these topologies in the three-dimensional case.

Definition
Given a Riemannian metric $g$, the scalar curvature Scal is defined as the trace of the Ricci curvature tensor with respect to the metric:

 * $$\operatorname{Scal} = \operatorname{tr}_g \operatorname{Ric}.$$

The scalar curvature cannot be computed directly from the Ricci curvature since the latter is a (0,2)-tensor field; the metric must be used to raise an index to obtain a (1,1)-tensor field in order to take the trace. In terms of local coordinates one can write, using the Einstein notation convention, that:
 * $$\operatorname{Scal} = g^{ij}R_{ij}$$

where $R_{ij} = Ric(∂_{i}, ∂_{j})$ are the components of the Ricci tensor in the coordinate basis, and where $g^{ij}$ are the inverse metric components, i.e. the components of the inverse of the matrix of metric components $g_{ij} = g(∂_{i}, ∂_{j})$. Based upon the Ricci curvature being a sum of sectional curvatures, it is possible to also express the scalar curvature as
 * $$\operatorname{Scal}(p)=\sum_{i\neq j}\operatorname{Sec}(e_i,e_j)$$

where $Sec$ denotes the sectional curvature and $e_{1}, ..., e_{n}$ is any orthonormal frame at $p$. By similar reasoning, the scalar curvature is twice the trace of the curvature operator. Alternatively, given the coordinate-based definition of Ricci curvature in terms of the Christoffel symbols, it is possible to express scalar curvature as

\operatorname{Scal} = g^{\mu \nu} \left({\Gamma^\lambda}_{\mu\nu,\lambda} - {\Gamma^\lambda}_{\mu\lambda,\nu} + {\Gamma^\sigma}_{\mu \nu}{\Gamma^\lambda}_{\lambda \sigma} - {\Gamma^\sigma}_{\mu \lambda}{\Gamma^\lambda}_{\nu \sigma}\right) $$ where $${\Gamma^\mu}_{\nu \lambda}$$ are the Christoffel symbols of the metric, and $${\Gamma^\mu}_{\nu \lambda,\sigma}$$ is the partial derivative of $${\Gamma^\mu}_{\nu \lambda}$$ in the σ-coordinate direction.

The above definitions are equally valid for a pseudo-Riemannian metric. The special case of Lorentzian metrics is significant in the mathematical theory of general relativity, where the scalar curvature and Ricci curvature are the fundamental terms in the Einstein field equation.

However, unlike the Riemann curvature tensor or the Ricci tensor, the scalar curvature cannot be defined for an arbitrary affine connection, for the reason that the trace of a (0,2)-tensor field is ill-defined. However, there are other generalizations of scalar curvature, including in Finsler geometry.

Traditional notation
In the context of tensor index notation, it is common to use the letter $R$ to represent three different things:
 * 1) the Riemann curvature tensor: $R_{ijk}^{l}$ or $R_{ijkl}$
 * 2) the Ricci tensor: $R_{ij}$
 * 3) the scalar curvature: $R$

These three are then distinguished from each other by their number of indices: the Riemann tensor has four indices, the Ricci tensor has two indices, and the Ricci scalar has zero indices. Other notations used for scalar curvature include $scal$, $κ$, $K$, $r$, $s$ or $S$, and $τ$.

Those not using an index notation usually reserve R for the full Riemann curvature tensor. Alternatively, in a coordinate-free notation one may use Riem for the Riemann tensor, Ric for the Ricci tensor and R for the scalar curvature.

Some authors instead define Ricci curvature and scalar curvature with a normalization factor, so that
 * $$R_{ij}=\frac{1}{n-1}g^{kl}R_{kijl}\text{ and }R=\frac{1}{n}g^{ij}R_{ij}.$$

The purpose of such a choice is that the Ricci and scalar curvatures become average values (rather than sums) of sectional curvatures.

Basic properties
It is a fundamental fact that the scalar curvature is invariant under isometries. To be precise, if $f$ is a diffeomorphism from a space $M$ to a space $N$, the latter being equipped with a (pseudo-)Riemannian metric $g$, then the scalar curvature of the pullback metric on $M$ equals the composition of the scalar curvature of $g$ with the map $f$. This amounts to the assertion that the scalar curvature is geometrically well-defined, independent of any choice of coordinate chart or local frame. More generally, as may be phrased in the language of homotheties, the effect of scaling the metric by a constant factor $c$ is to scale the scalar curvature by the inverse factor $c^{−1}$.

Furthermore, the scalar curvature is (up to an arbitrary choice of normalization factor) the only coordinate-independent function of the metric which, as evaluated at the center of a normal coordinate chart, is a polynomial in derivatives of the metric and has the above scaling property. This is one formulation of the Vermeil theorem.

Bianchi identity
As a direct consequence of the Bianchi identities, any (pseudo-)Riemannian metric has the property that
 * $$\frac{1}{2}\nabla_iR=g^{jk}\nabla_jR_{ki}.$$

This identity is called the contracted Bianchi identity. It has, as an almost immediate consequence, the Schur lemma stating that if the Ricci tensor is pointwise a multiple of the metric, then the metric must be Einstein (unless the dimension is two). Moreover, this says that (except in two dimensions) a metric is Einstein if and only if the Ricci tensor and scalar curvature are related by
 * $$R_{ij}=\frac{1}{n}Rg_{ij},$$

where $n$ denotes the dimension. The contracted Bianchi identity is also fundamental in the mathematics of general relativity, since it identifies the Einstein tensor as a fundamental quantity.

Ricci decomposition
Given a (pseudo-)Riemannian metric $g$ on a space of dimension $n$, the scalar curvature part of the Riemann curvature tensor is the (0,4)-tensor field
 * $$\frac{1}{n(n-1)}R(g_{il}g_{jk}-g_{ik}g_{jl}).$$

(This follows the convention that $R_{ijkl} = g_{lp}∂_{i}Γ_{jk}^{p} − ...$.) This tensor is significant as part of the Ricci decomposition; it is orthogonal to the difference between the Riemann tensor and itself. The other two parts of the Ricci decomposition correspond to the components of the Ricci curvature which do not contribute to scalar curvature, and to the Weyl tensor, which is the part of the Riemann tensor which does not contribute to the Ricci curvature. Put differently, the above tensor field is the only part of the Riemann curvature tensor which contributes to the scalar curvature; the other parts are orthogonal to it and make no such contribution. There is also a Ricci decomposition for the curvature of a Kähler metric.

Basic formulas
The scalar curvature of a conformally changed metric can be computed:
 * $$R(e^{2f}g)=e^{-2f}\Big(R(g)-2(n-1)\Delta^gf-(n-2)(n-1)g(df,df)\Big),$$

using the convention $Δ = g^{ij }∇_{i}∇_{j}$ for the Laplace–Beltrami operator. Alternatively,
 * $$R(\psi^{4/(n-2)}g)=-\frac{4\frac{n-1}{n-2}\Delta^g\psi-R(g)\psi}{\psi^{\frac{n+2}{n-2}}}.$$

Under an infinitesimal change of the underlying metric, one has
 * $$\frac{\partial R}{\partial t}=-\Delta^g\left(g^{ij}\frac{\partial g_{ij}}{\partial t}\right)+\left(\nabla_k\nabla_l\frac{\partial g_{ij}}{\partial t}-R_{kl}\frac{\partial g_{ij}}{\partial t}\right)g^{ik}g^{jl}.$$

This shows in particular that the principal symbol of the differential operator which sends a metric to its scalar curvature is given by
 * $$(\xi_i,h_{ij})\mapsto -g(\xi,\xi)g^{ij}h_{ij}+h_{ij}\xi^i\xi^j.$$

Furthermore the adjoint of the linearized scalar curvature operator is
 * $$f\mapsto \nabla_i\nabla_jf-(\Delta f)g_{ij}-fR_{ij},$$

and it is an overdetermined elliptic operator in the case of a Riemannian metric. It is a straightforward consequence of the first variation formulas that, to first order, a Ricci-flat Riemannian metric on a closed manifold cannot be deformed so as to have either positive or negative scalar curvature. Also to first order, an Einstein metric on a closed manifold cannot be deformed under a volume normalization so as to increase or decrease scalar curvature.

Relation between volume and Riemannian scalar curvature
When the scalar curvature is positive at a point, the volume of a small geodesic ball about the point has smaller volume than a ball of the same radius in Euclidean space. On the other hand, when the scalar curvature is negative at a point, the volume of a small ball is larger than it would be in Euclidean space.

This can be made more quantitative, in order to characterize the precise value of the scalar curvature S at a point p of a Riemannian n-manifold $$(M,g)$$. Namely, the ratio of the n-dimensional volume of a ball of radius ε in the manifold to that of a corresponding ball in Euclidean space is given, for small ε, by

\frac{\operatorname{Vol}(B_\varepsilon(p) \subset M)}{\operatorname{Vol}\left(B_\varepsilon(0)\subset {\mathbb R}^n\right)} = 1 - \frac{S}{6(n + 2)}\varepsilon^2 + O\left(\varepsilon^3\right). $$

Thus, the second derivative of this ratio, evaluated at radius ε = 0, is exactly minus the scalar curvature divided by 3(n + 2).

Boundaries of these balls are (n − 1)-dimensional spheres of radius $$\varepsilon$$; their hypersurface measures ("areas") satisfy the following equation:

 * $$\frac{\operatorname{Area} (\partial B_\varepsilon(p) \subset M)}{\operatorname{Area}(\partial B_\varepsilon(0)\subset {\mathbb R}^n)} = 1 - \frac{S}{6n} \varepsilon^2 + O\left(\varepsilon^3\right).$$

These expansions generalize certain characterizations of Gaussian curvature from dimension two to higher dimensions.

Surfaces
In two dimensions, scalar curvature is exactly twice the Gaussian curvature. For an embedded surface in Euclidean space R3, this means that

 * $$ S = \frac{2}{\rho_1\rho_2}\,$$

where $$\rho_1,\,\rho_2$$ are the principal radii of the surface. For example, the scalar curvature of the 2-sphere of radius r is equal to 2/r2.

The 2-dimensional Riemann curvature tensor has only one independent component, and it can be expressed in terms of the scalar curvature and metric area form. Namely, in any coordinate system, one has

 * $$2R_{1212} \,= S \det (g_{ij}) = S\left[g_{11}g_{22} - (g_{12})^2\right].$$

Space forms
A space form is by definition a Riemannian manifold with constant sectional curvature. Space forms are locally isometric to one of the following types: Euclidean space:The Riemann tensor of an n-dimensional Euclidean space vanishes identically, so the scalar curvature does as well. n-spheres:The sectional curvature of an n-sphere of radius r is K = 1/r2. Hence the scalar curvature is S = n(n − 1)/r2. Hyperbolic space:By the hyperboloid model, an n-dimensional hyperbolic space can be identified with the subset of (n + 1)-dimensional Minkowski space
 * $x_0^2 - x_1^2 - \cdots - x_n^2 = r^2,\quad x_0 > 0.$

The parameter r is a geometrical invariant of the hyperbolic space, and the sectional curvature is K = −1/r2. The scalar curvature is thus S = −n(n − 1)/r2. The scalar curvature is also constant when given a Kähler metric of constant holomorphic sectional curvature.

Products
The scalar curvature of a product M × N of Riemannian manifolds is the sum of the scalar curvatures of M and N. For example, for any smooth closed manifold M, M × S2 has a metric of positive scalar curvature, simply by taking the 2-sphere to be small compared to M (so that its curvature is large). This example might suggest that scalar curvature has little relation to the global geometry of a manifold. In fact, it does have some global significance, as discussed below.

In both mathematics and general relativity, warped product metrics are an important source of examples. For example, the general Robertson–Walker spacetime, important to cosmology, is the Lorentzian metric
 * $$-dt^2+f(t)^2 g$$

on $(a, b) × M$, where $g$ is a constant-curvature Riemannian metric on a three-dimensional manifold $M$. The scalar curvature of the Robertson–Walker metric is given by
 * $$6\frac{f'(t)^2+f(t)f''(t)+k}{f(t)^2},$$

where $k$ is the constant curvature of $g$.

Scalar-flat spaces
It is automatic that any Ricci-flat manifold has zero scalar curvature; the best-known spaces in this class are the Calabi–Yau manifolds. In the pseudo-Riemannian context, this also includes the Schwarzschild spacetime and Kerr spacetime.

There are metrics with zero scalar curvature but nonvanishing Ricci curvature. For example, there is a complete Riemannian metric on the tautological line bundle over real projective space, constructed as a warped product metric, which has zero scalar curvature but nonzero Ricci curvature. This may also be viewed as a rotationally symmetric Riemannian metric of zero scalar curvature on the cylinder $R × S^{n}$.

Yamabe problem
The Yamabe problem was resolved in 1984 by the combination of results found by Hidehiko Yamabe, Neil Trudinger, Thierry Aubin, and Richard Schoen. They proved that every smooth Riemannian metric on a closed manifold can be multiplied by some smooth positive function to obtain a metric with constant scalar curvature. In other words, every Riemannian metric on a closed manifold is conformal to one with constant scalar curvature.

Riemannian metrics of positive scalar curvature
For a closed Riemannian 2-manifold M, the scalar curvature has a clear relation to the topology of M, expressed by the Gauss–Bonnet theorem: the total scalar curvature of M is equal to 4$\pi$ times the Euler characteristic of M. For example, the only closed surfaces with metrics of positive scalar curvature are those with positive Euler characteristic: the sphere S2 and RP2. Also, those two surfaces have no metrics with scalar curvature ≤ 0.

Nonexistence results
In the 1960s, André Lichnerowicz found that on a spin manifold, the difference between the square of the Dirac operator and the tensor Laplacian (as defined on spinor fields) is given exactly by one-quarter of the scalar curvature. This is a fundamental example of a Weitzenböck formula. As a consequence, if a Riemannian metric on a closed manifold has positive scalar curvature, then there can exist no harmonic spinors. It is then a consequence of the Atiyah–Singer index theorem that, for any closed spin manifold with dimension divisible by four and of positive scalar curvature, the Â genus must vanish. This is a purely topological obstruction to the existence of Riemannian metrics with positive scalar curvature.

Lichnerowicz's argument using the Dirac operator can be "twisted" by an auxiliary vector bundle, with the effect of only introducing one extra term into the Lichnerowicz formula. Then, following the same analysis as above except using the families version of the index theorem and a refined version of the Â genus known as the α-genus, Nigel Hitchin proved that in certain dimensions there are exotic spheres which do not have any Riemannian metrics of positive scalar curvature. Gromov and Lawson later extensively employed these variants of Lichnerowicz's work. One of their resulting theorems introduces the homotopy-theoretic notion of enlargeability and says that an enlargeable spin manifold cannot have a Riemannian metric of positive scalar curvature. As a corollary, a closed manifold with a Riemannian metric of nonpositive curvature, such as a torus, has no metric with positive scalar curvature. Gromov and Lawson's various results on nonexistence of Riemannian metrics with positive scalar curvature support a conjecture on the vanishing of a wide variety of topological invariants of any closed spin manifold with positive scalar curvature. This (in a precise formulation) in turn would be a special case of the strong Novikov conjecture for the fundamental group, which deals with the K-theory of C*-algebras. This in turn is a special case of the Baum–Connes conjecture for the fundamental group.

In the special case of four-dimensional manifolds, the Seiberg–Witten equations have been usefully applied to the study of scalar curvature. Similarly to Lichnerowicz's analysis, the key is an application of the maximum principle to prove that solutions to the Seiberg–Witten equations must be trivial when scalar curvature is positive. Also in analogy to Lichnerowicz's work, index theorems can guarantee the existence of nontrivial solutions of the equations. Such analysis provides new criteria for nonexistence of metrics of positive scalar curvature. Claude LeBrun pursued such ideas in a number of papers.

Existence results
By contrast to the above nonexistence results, Lawson and Yau constructed Riemannian metrics of positive scalar curvature from a wide class of nonabelian effective group actions.

Later, Schoen–Yau and Gromov–Lawson (using different techniques) proved the fundamental result that existence of Riemannian metrics of positive scalar curvature is preserved by topological surgery in codimension at least three, and in particular is preserved by the connected sum. This establishes the existence of such metrics on a wide variety of manifolds. For example, it immediately shows that the connected sum of an arbitrary number of copies of spherical space forms and generalized cylinders $S^{m} × S^{n}$ has a Riemannian metric of positive scalar curvature. Grigori Perelman's construction of Ricci flow with surgery has, as an immediate corollary, the converse in the three-dimensional case: a closed orientable 3-manifold with a Riemannian metric of positive scalar curvature must be such a connected sum.

Based upon the surgery allowed by the Gromov–Lawson and Schoen–Yau construction, Gromov and Lawson observed that the h-cobordism theorem and analysis of the cobordism ring can be directly applied. They proved that, in dimension greater than four, any non-spin simply connected closed manifold has a Riemannian metric of positive scalar curvature. Stephan Stolz completed the existence theory for simply-connected closed manifolds in dimension greater than four, showing that as long as the α-genus is zero, then there is a Riemannian metric of positive scalar curvature.

According to these results, for closed manifolds, the existence of Riemannian metrics of positive scalar curvature is completely settled in the three-dimensional case and in the case of simply-connected manifolds of dimension greater than four.

Kazdan and Warner's trichotomy theorem
The sign of the scalar curvature has a weaker relation to topology in higher dimensions. Given a smooth closed manifold M of dimension at least 3, Kazdan and Warner solved the prescribed scalar curvature problem, describing which smooth functions on M arise as the scalar curvature of some Riemannian metric on M. Namely, M must be of exactly one of the following three types:
 * 1) Every function on M is the scalar curvature of some metric on M.
 * 2) A function on M is the scalar curvature of some metric on M if and only if it is either identically zero or negative somewhere.
 * 3) A function on M is the scalar curvature of some metric on M if and only if it is negative somewhere.

Thus every manifold of dimension at least 3 has a metric with negative scalar curvature, in fact of constant negative scalar curvature. Kazdan–Warner's result focuses attention on the question of which manifolds have a metric with positive scalar curvature, that being equivalent to property (1). The borderline case (2) can be described as the class of manifolds with a strongly scalar-flat metric, meaning a metric with scalar curvature zero such that M has no metric with positive scalar curvature.

Akito Futaki showed that strongly scalar-flat metrics (as defined above) are extremely special. For a simply connected Riemannian manifold M of dimension at least 5 which is strongly scalar-flat, M must be a product of Riemannian manifolds with holonomy group SU(n) (Calabi–Yau manifolds), Sp(n) (hyperkähler manifolds), or Spin(7). In particular, these metrics are Ricci-flat, not just scalar-flat. Conversely, there are examples of manifolds with these holonomy groups, such as the K3 surface, which are spin and have nonzero α-invariant, hence are strongly scalar-flat.

title: Scalar curvature


text:
Gaussian curvature

In differential geometry, the Gaussian curvature or Gauss curvature $Κ$ of a smooth surface in three-dimensional space at a point is the product of the principal curvatures, $κ_{1}$ and $κ_{2}$, at the given point: $$ K = \kappa_1 \kappa_2.$$ The Gaussian radius of curvature is the reciprocal of $Κ$. For example, a sphere of radius $r$ has Gaussian curvature $1⁄r^{2}$ everywhere, and a flat plane and a cylinder have Gaussian curvature zero everywhere. The Gaussian curvature can also be negative, as in the case of a hyperboloid or the inside of a torus.

Gaussian curvature is an intrinsic measure of curvature, depending only on distances that are measured “within” or along the surface, not on the way it is isometrically embedded in Euclidean space. This is the content of the Theorema egregium.

Gaussian curvature is named after Carl Friedrich Gauss, who published the Theorema egregium in 1827.

Informal definition

At any point on a surface, we can find a normal vector that is at right angles to the surface; planes containing the normal vector are called normal planes. The intersection of a normal plane and the surface will form a curve called a normal section and the curvature of this curve is the normal curvature. For most points on most “smooth” surfaces, different normal sections will have different curvatures; the maximum and minimum values of these are called the principal curvatures, call these $κ_{1}$, $κ_{2}$. The Gaussian curvature is the product of the two principal curvatures $Κ = κ_{1}κ_{2}$.

The sign of the Gaussian curvature can be used to characterise the surface.
 * If both principal curvatures are of the same sign: $κ_{1}κ_{2} > 0$, then the Gaussian curvature is positive and the surface is said to have an elliptic point. At such points, the surface will be dome like, locally lying on one side of its tangent plane. All sectional curvatures will have the same sign.
 * If the principal curvatures have different signs: $κ_{1}κ_{2} < 0$, then the Gaussian curvature is negative and the surface is said to have a hyperbolic or saddle point. At such points, the surface will be saddle shaped. Because one principal curvature is negative, one is positive, and the normal curvature varies continuously if you rotate a plane orthogonal to the surface around the normal to the surface in two directions, the normal curvatures will be zero giving the asymptotic curves for that point.
 * If one of the principal curvatures is zero: $κ_{1}κ_{2} = 0$, the Gaussian curvature is zero and the surface is said to have a parabolic point.

Most surfaces will contain regions of positive Gaussian curvature (elliptical points) and regions of negative Gaussian curvature separated by a curve of points with zero Gaussian curvature called a parabolic line.

Relation to geometries
When a surface has a constant zero Gaussian curvature, then it is a developable surface and the geometry of the surface is Euclidean geometry.

When a surface has a constant positive Gaussian curvature, then the geometry of the surface is spherical geometry. Spheres and patches of spheres have this geometry, but there exist other examples as well, such as the lemon / American football.

When a surface has a constant negative Gaussian curvature, then it is a pseudospherical surface and the geometry of the surface is hyperbolic geometry.

Relation to principal curvatures
The two principal curvatures at a given point of a surface are the eigenvalues of the shape operator at the point. They measure how the surface bends by different amounts in different directions from that point. We represent the surface by the implicit function theorem as the graph of a function, $f$, of two variables, in such a way that the point $p$ is a critical point, that is, the gradient of $f$ vanishes (this can always be attained by a suitable rigid motion). Then the Gaussian curvature of the surface at $p$ is the determinant of the Hessian matrix of $f$ (being the product of the eigenvalues of the Hessian). (Recall that the Hessian is the 2×2 matrix of second derivatives.) This definition allows one immediately to grasp the distinction between a cup/cap versus a saddle point.

Alternative definitions
It is also given by $$K = \frac{\bigl\langle (\nabla_2 \nabla_1 - \nabla_1 \nabla_2)\mathbf{e}_1, \mathbf{e}_2\bigr\rangle}{\det g},$$ where $∇_{i} = ∇_{e_{i}}|undefined$ is the covariant derivative and $g$ is the metric tensor.

At a point $p$ on a regular surface in $R^{3}$, the Gaussian curvature is also given by $$K(\mathbf{p}) = \det S(\mathbf{p}),$$ where $S$ is the shape operator.

A useful formula for the Gaussian curvature is Liouville's equation in terms of the Laplacian in isothermal coordinates.

Total curvature

The surface integral of the Gaussian curvature over some region of a surface is called the total curvature. The total curvature of a geodesic triangle equals the deviation of the sum of its angles from $\pi$. The sum of the angles of a triangle on a surface of positive curvature will exceed π, while the sum of the angles of a triangle on a surface of negative curvature will be less than π. On a surface of zero curvature, such as the Euclidean plane, the angles will sum to precisely π radians. $$\sum_{i=1}^3 \theta_i = \pi + \iint_T K \,dA.$$ A more general result is the Gauss–Bonnet theorem.

Theorema egregium
Gauss's Theorema egregium (Latin: "remarkable theorem") states that Gaussian curvature of a surface can be determined from the measurements of length on the surface itself. In fact, it can be found given the full knowledge of the first fundamental form and expressed via the first fundamental form and its partial derivatives of first and second order. Equivalently, the determinant of the second fundamental form of a surface in $R^{3}$ can be so expressed. The "remarkable", and surprising, feature of this theorem is that although the definition of the Gaussian curvature of a surface $S$ in $R^{3}$ certainly depends on the way in which the surface is located in space, the end result, the Gaussian curvature itself, is determined by the intrinsic metric of the surface without any further reference to the ambient space: it is an intrinsic invariant. In particular, the Gaussian curvature is invariant under isometric deformations of the surface.

In contemporary differential geometry, a "surface", viewed abstractly, is a two-dimensional differentiable manifold. To connect this point of view with the classical theory of surfaces, such an abstract surface is embedded into $R^{3}$ and endowed with the Riemannian metric given by the first fundamental form. Suppose that the image of the embedding is a surface $S$ in $R^{3}$. A local isometry is a diffeomorphism $f : U → V$ between open regions of $R^{3}$ whose restriction to $S ∩ U$ is an isometry onto its image. Theorema egregium is then stated as follows:

$$

For example, the Gaussian curvature of a cylindrical tube is zero, the same as for the "unrolled" tube (which is flat). On the other hand, since a sphere of radius $R$ has constant positive curvature $R^{3}$ and a flat plane has constant curvature 0, these two surfaces are not isometric, not even locally. Thus any planar representation of even a small part of a sphere must distort the distances. Therefore, no cartographic projection is perfect.

Gauss–Bonnet theorem
The Gauss–Bonnet theorem relates the total curvature of a surface to its Euler characteristic and provides an important link between local geometric properties and global topological properties.
 * $$\int_M K\,dA+\int_{\partial M}k_g\,ds=2\pi\chi(M), \, $$

Surfaces of constant curvature

 * Minding's theorem (1839) states that all surfaces with the same constant curvature $K$ are locally isometric. A consequence of Minding's theorem is that any surface whose curvature is identically zero can be constructed by bending some plane region. Such surfaces are called developable surfaces. Minding also raised the question of whether a closed surface with constant positive curvature is necessarily rigid.
 * Liebmann's theorem (1900) answered Minding's question. The only regular (of class $R^{−2}$) closed surfaces in $C^{2}$ with constant positive Gaussian curvature are spheres. If a sphere is deformed, it does not remain a sphere, proving that a sphere is rigid. A standard proof uses Hilbert's lemma that non-umbilical points of extreme principal curvature have non-positive Gaussian curvature.
 * Hilbert's theorem (1901) states that there exists no complete analytic (class $R^{3}$) regular surface in $C^{ω}$ of constant negative Gaussian curvature. In fact, the conclusion also holds for surfaces of class $R^{3}$ immersed in $C^{2}$, but breaks down for $R^{3}$-surfaces. The pseudosphere has constant negative Gaussian curvature except at its boundary circle, where the gaussian curvature is not defined.

There are other surfaces which have constant positive Gaussian curvature. Manfredo do Carmo considers surfaces of revolution $$(\phi(v) \cos(u), \phi(v) \sin(u), \psi(v))$$ where $$\phi(v) = C \cos v$$, and $ \psi(v) = \int_0^v \sqrt{1-C^2 \sin^2 v'}\ dv'$  (an incomplete Elliptic integral of the second kind). These surfaces all have constant Gaussian curvature of 1, but, for $$C\ne 1$$ either have a boundary or a singular point. do Carmo also gives three different examples of surface with constant negative Gaussian curvature, one of which is pseudosphere.

There are many other possible bounded surfaces with constant Gaussian curvature. Whilst the sphere is rigid and can not be bent using an isometry, if a small region removed, or even a cut along a small segment, then the resulting surface can be bent. Such bending preserves Gaussian curvature so any such bending of a sphere with a region removed will also have constant Gaussian curvature.

Alternative formulas
K = -\frac{ \begin{vmatrix} H(F) & \nabla F^{\mathsf T} \\ \nabla F & 0 \end{vmatrix} }{ |\nabla F|^4 } =-\frac{ \begin{vmatrix} F_{xx} & F_{xy} & F_{xz} & F_x \\ F_{xy} & F_{yy} & F_{yz} & F_y \\ F_{xz} & F_{yz} & F_{zz} & F_z \\ F_{x} & F_{y} & F_{z} & 0 \\ \end{vmatrix} }{ |\nabla F|^4 } $$
 * Gaussian curvature of a surface in $C^{1}$ can be expressed as the ratio of the determinants of the second and first fundamental forms $R^{3}$ and $II$: $$K = \frac{\det(\mathrm{I\!I})}{\det(\mathrm I)} = \frac{LN-M^2}{EG-F^2}.$$
 * The Brioschi formula (after Francesco Brioschi) gives Gaussian curvature solely in terms of the first fundamental form: $$ K =\frac{\begin{vmatrix} -\frac{1}{2}E_{vv} + F_{uv} - \frac{1}{2}G_{uu} & \frac{1}{2} E_u & F_u-\frac{1}{2}E_v\\F_v-\frac{1}{2}G_u & E & F\\\frac{1}{2}G_v & F & G \end{vmatrix} - \begin{vmatrix} 0 & \frac{1}{2} E_v & \frac{1}{2} G_u\\\frac{1}{2} E_v & E & F\\\frac{1}{2}G_u & F & G \end{vmatrix}}{\left(EG - F^2\right)^2} $$
 * For an orthogonal parametrization ($I$), Gaussian curvature is: $$K = -\frac{1}{2\sqrt{EG}}\left(\frac{\partial}{\partial u}\frac{G_u}{\sqrt{EG}} + \frac{\partial}{\partial v} \frac{E_v}{\sqrt{EG}}\right).$$
 * For a surface described as graph of a function $F = 0$, Gaussian curvature is: $$K = \frac{F_{xx}\cdot F_{yy}- F_{xy}^2}{\left(1+F_x^2+ F_y^2\right)^2}$$
 * For an implicitly defined surface, $z = F(x,y)$, the Gaussian curvature can be expressed in terms of the gradient $F(x,y,z) = 0$ and Hessian matrix $∇F$: $$
 * For a surface with metric conformal to the Euclidean one, so $H(F)$ and $F = 0$, the Gauss curvature is given by ($E = G = e^{σ}$ being the usual Laplace operator): $$ K = -\frac{1}{2e^\sigma}\Delta \sigma.$$
 * Gaussian curvature is the limiting difference between the circumference of a geodesic circle and a circle in the plane: $$ K = \lim_{r\to 0^+} 3\frac{2\pi r-C(r)}{\pi r^3}$$
 * Gaussian curvature is the limiting difference between the area of a geodesic disk and a disk in the plane: $$K = \lim_{r\to 0^+}12\frac{\pi r^2-A(r)}{\pi r^4 } $$
 * Gaussian curvature may be expressed with the Christoffel symbols: $$K = -\frac{1}{E} \left( \frac{\partial}{\partial u}\Gamma_{12}^2 - \frac{\partial}{\partial v}\Gamma_{11}^2 + \Gamma_{12}^1\Gamma_{11}^2 - \Gamma_{11}^1\Gamma_{12}^2 + \Gamma_{12}^2\Gamma_{12}^2 - \Gamma_{11}^2\Gamma_{22}^2\right)$$

title: Gaussian curvature


text:
Wu

Wu may refer to:

Places

 * Wu (region), a region roughly corresponding to the territory of Wuyue
 * Wu Chinese, a subgroup of Chinese languages now spoken in the Wu region
 * Wuyue culture (吳越文化), a regional Chinese culture in the Wu region
 * Wu (state) (*), a kingdom during the Spring and Autumn period 771–476 BCE
 * Suzhou or Wu, its eponymous capital
 * Wu County, a former county in Suzhou
 * Eastern Wu or Sun Wu, one of the Three Kingdoms in 184/220–280 CE
 * Li Zitong (吳帝, died 622), who declared a brief Wu dynasty during the Sui–Tang interregnum in 619–620 CE
 * Wu (Ten Kingdoms), one of the ten kingdoms during the Five Dynasties and Ten Kingdoms period 907–960 CE
 * Wuyue, another of the ten kingdoms during the Five Dynasties and Ten Kingdoms period 907–960 CE
 * Wu River (disambiguation), various rivers in China

Language

 * Wu Chinese, a group of Sinitic languages that includes Shanghaiese

People

 * Wu (surname) (or Woo) (吳), several different Chinese surnames
 * Wu (surname 伍)
 * Wu (surname 武)
 * Chinese rulers
 * Emperor Wu (disambiguation)
 * King Wu (disambiguation)
 * Duke Wu (disambiguation)
 * Wu Zetian (624–705), also known as Empress Wu
 * Wu, nickname of the singer of alternative rock band This Et Al

Religion

 * Wu (awareness), a concept of awareness, consciousness or enlightenment in the Chinese folk religion
 * Wú (negative), Zhaozhou's response to the question: "Does a dog have the Buddha-nature?"
 * Wu (shaman), shaman in Chinese folk religion

Arts and entertainment

 * "Doctor Wu", a song on the 1975 album Katy Lied by the band Steely Dan
 * Mr. Wu (1919 film), a 1919 British drama film based on the 1913 play
 * Mr. Wu (1927 film), a 1927 American silent movie starring Lon Chaney

Characters

 * Mr. Wu, a character referred to in several songs of the 1930s and 1940s by George Formby
 * Mr. Wu, a character from the HBO TV series Deadwood
 * Sensei Wu, a character in Ninjago
 * Sgt. Wu, a character from the TV series Grimm
 * Marcy Wu, a character from the TV series Amphibia
 * Madame Wu, a character in the Simpsons episode "Goo Goo Gai Pan"

title: Wu


text:
Talk:R

Pronunciation
I notice that there are no entries for the "vowel sounds" that are ways in which this letter is pronounced in British English (all right, Australian too) and in German. German in particular has one nasty pronunciation shown in IPA by the upside-down letter "a". Also, the the German Theatrical Pronunciation, i.e., "trilled", should be noted. And the "r-colored" vowels should also have a separate listing at the pronunciation table at the bottom. --Sobolewski 16:41, 24 January 2006 (UTC)

 * The nasty German pronunciation, as you put it, sounds much the same as the British English pronunciation of the "er" in "letter" or "teacher".--Unoffensive text or character 07:58, 25 July 2006 (UTC)

 * I've added the German alveolar and uvular trills, and AmE retroflex approximant. Incidentally, given that Arabic, Armenian, Greek, Hebrew, Hindi and Japanese are not usually written in the Roman script, how is it meaningful for them to appear in this table? Hairy Dude 00:28, 28 May 2006 (UTC)

In the pronunciation grid, French and Dutch are in the same row: this is not correct? The French 'R' is a voiced velar fricative! - Katleen, Belgium

I too have a few comments on the pronunciation of the phoneme /r/ in Dutch. It is very often realised as an alveolar trill or tap (most people I know use these). By some Netherlands Dutch speakers, an alveolar approximant is used in some contexts. An uvular trill is not uncommon either, but much more common are uvular approximants and uvular voiced fricatives. A pre-velar bunched approximant (comparable to a retroflex) is used word-final in Netherlands Dutch. See Beverley Collins & Inger M. Mees, The Phonetics of English and Dutch (2003:pp.199-200) for more information.-- Lambik (Belgium) 16:53, 4 June 2006 (UTC)

Since this is an English article, I think the IPA pronunciation in the beginning of the article should be correct for English. Few dialects use the alveolar trill, as is in the given pronunciation. Please use the 180-degree rotated r. —Preceding unsigned comment added by 99.38.146.6 (talk) 21:47, 11 March 2011 (UTC)

thermal resistance
R is also thermal resistance —Preceding unsigned comment added by 165.230.19.153 (talk • contribs)
 * I'll add it in.--58.104.75.186 07:20, 16 July 2006 (UTC)

Unicode problem
The character ℝ (double-struck R) in the article appears on my computer as a confused box. This could be because I have only five obscure fonts that contain it, but shouldn’t the Unicode template (used here) handle that? I see the character as I type this, but after I save it, it’s a confused box. —Frungi 18:10, 21 November 2006 (UTC)
 * That’s odd, I’m not having the problem anymore, even though no change has been made to this article or the template. —Frungi 02:25, 27 November 2006 (UTC)

amino acid group
in biochemistry is an R group not also the changeable group on an amino acid, attached to the central carbon atom?--Max Randor 21:10, 7 January 2007 (UTC)

Audio
The uvular trill sounds particularly nasty and the voiced uvular fricative is much to harsh. Both need to be taken down a few notches. The voiced uvular fricative should be almost frictionless.

Also the Japanese R (Alveolar lateral flap) needs a better (and preferably ogg) audio file. I wish I could contribute it myself but I'm having problems with my mic input. 74.129.182.66 21:31, 4 March 2007 (UTC)

R in Norwegian
In the table in the section Various phonetic realizations of R, I've added Norwegian to the "Alveolar trill" row, since this is the regular pronunciation in dialects not using the uvular trill. Also, I've removed Norwegian from the "Retroflex flap" row -- the retroflex flap is used in (dialectal) Norwegian, but as a realization of L, not R (see the article Retroflex flap). —Preceding unsigned comment added by OMHalck (talk • contribs) 11:11, 15 September 2007 (UTC)

"Various phonetic realizations of R"
I have a couple of issues with this section. Firstly, what is the phonological meaning of a 'weaker' pronunciation of a sound. Secondly, was the 'r' sound really pronounced by English English speakers as an alveolar trill as recently as the late nineteenth and early twentieth centuries? I associate this pronunciation as being mostly Scottish even a century ago. 142.157.208.85 (talk) 10:57, 13 December 2007 (UTC)

Rolled R
What about rolled r's ([ɛr]) such as in Icelandic? --Stefán Örvarr Sigmundsson (talk) 03:56, 5 August 2008 (UTC)

Egyptian R
The head sign on this page has a phonetic value of "tp." The real R is  97.127.34.184 (talk) 21:05, 15 July 2009 (UTC)
 * The article isn't concerned with the phonetic value of the hieroglyph, but with which one influenced the shape of the Semitic letter that influenced the Greek letter that gave us the Latin letter R. Anomie⚔ 23:59, 15 July 2009 (UTC)
 * That's fine, as long as tp is not labeled as being actually an R, as it was at the time. 97.127.53.199 (talk) 16:55, 24 September 2009 (UTC)

The head sign is transliterated as 'tp'. It does not refer to its phonetic pronunciation. —Preceding unsigned comment added by 111.240.50.144 (talk) 15:05, 26 March 2011 (UTC)

Xhosa R
I think there should be a mention of the Xhosa pronunciation of the letter r, which is almost similar to the Hebrew CH. —Preceding unsigned comment added by Alirie (talk • contribs) 03:49, 21 November 2009 (UTC)

Fictional R
Alphablock R (female character) does the R sound. The R suits her name cos she's acts and looks like a pirate. —Preceding unsigned comment added by 90.206.5.60 (talk) 16:29, 11 August 2010 (UTC)

most varieties of English
Both the alveolar approximant and the retroflex approximant are now attributed to "most varieties" of English, hm. Curiously the "alveolar" ogg file sounds retroflex to me, and the "retroflex" ogg does not! —Tamfang (talk) 01:23, 3 April 2011 (UTC)

templates?
Should [[Media:...]] be replaced with Listen templates to embed the sound files into the table? I just clicked on "listen" and my browser downloaded a file. Typically, I don't want anything downloaded to my computer. I just want to hear the file without having to open up more software. Any opinions? I haven't looked at other similar articles to see what they've done. –  Ker αun oςc op ia◁ galaxies  20:20, 14 April 2011 (UTC)

In the Related letters"
In the "Related letters and other similar characters" section, is the Cyrillic letter Er actually similar to R? I am asking about the similarity. Speling12345 (talk) 3:55, 13 December 2013 (UTC)

Sagar ubale
SAGAR UBALE REPUBLICAN PARTY OF INDIA(A) SHAHAPUR TALUKA VICE-PRESIDENT(YUTH) Rohan7798 (talk) 14:15, 18 November 2016 (UTC)

R
I just want my sandbox, people?!?-Katavra Sunburst Katavra Sunburst (talk) 01:13, 23 March 2018 (UTC)

Special character rendering support for R Rotunda
Hi, I don’t know how to do this myself, or I would. Whoever knows how, just go ahead, I’ll figure it out when I have more time. This page doesn’t render properly in all the places the “R Rotunda” character (Unicode U+A75A) is typed, so this page needs the template that says “this page has special characters ...” (I’m unclear on the actual wording.) Thanks!

RobbertMacGreighgor (talk) 07:00, 15 August 2018 (UTC)

Cursive R
The section on cursive doesn't actually discuss the usage of a separate glyph for r in modern cursive fonts. However, there is definitely a different glyph used in several cursive typefaces, it is especially useful because of the difficulty of joining a `normal' r to a cursive s and several similar joins. It would be good to have some discussion of this. — Preceding unsigned comment added by 120.18.115.123 (talk) 07:34, 30 April 2019 (UTC)

"ℛ" listed at Redirects for discussion
An editor has asked for a discussion to address the redirect ℛ. Please participate in the redirect discussion if you wish to do so. 1234qwer1234qwer4 (talk) 18:18, 31 March 2020 (UTC)

Rhoticity in the lead?
Should we say something about rhoticity in the lead. We have a section on pronunciation. If yes, can someone write it? 𝕁𝕄𝔽 (talk) 14:31, 3 March 2024 (UTC)

 * I think it's pretty important, yes. Many graphemics articles wouldn't have a lot to talk about if they didn't discuss the associated phones and phonemes in spoken languages. Remsense  诉  14:41, 3 March 2024 (UTC)

Indian rupee sign
Should it be listed as a descendent or not? it's derived from the Latin letter R and also has inspiration from the Devanagari symbol ra. please make your opinion known before I edit it.

; P Daisytheduck (talk) 22:53, 15 May 2024 (UTC)

title: Talk:R


text:
Hélène Dutrieu

Hélène Dutrieu (10 July 1877 – 26 June 1961), was a Belgian cycling world champion, stunt cyclist, stunt motorcyclist, automobile racer, stunt driver, pioneer pilot, wartime ambulance driver, and director of a military hospital.

Biography
Hélène Marguerite Dutrieu was born on 10 July 1877 in Tournai, Belgium, the daughter of a Belgian Army officer. The family later moved to Lille in northern France. She left school at the age of 14 to earn a living.

Cycling success
Hélène Dutrieu began her cycle racing career inspired by her older brother Eugène – a professional cyclist. She became a professional track cyclist racing for the Simpson Lever Chain team. In 1893 she gained the women's world record for distance cycled in one hour. In 1896 and 1897 she won the women's speed track cycling world championship in Ostend, Belgium, but lost her title in 1898 to Louise Roger. In November 1896 she won the Course de 12 Jours (12-day race) in the Royal Aquarium, London, England, and in August 1898 she won the Grand Prix d’Europe (Grand Prix of Europe). Leopold II of Belgium awarded Dutrieu the Cross of St André with diamonds in honour of her cycling success. She later became a motorcycle and automobile racer.

Stuntwoman
She began performing in variety shows as a cycling speciality act and in July 1903 she cycled a loop inside a vertical track at the Eldorado in Marseille, France. She also invented her own stunt, "La Flèche Humaine" ("The Human Arrow"), which was a c. 15 m jump with a bicycle. In September 1903 she appeared at l'Olympia, Paris. She also performed in London (Crystal Palace), Berlin and other main cities. Later, she moved on to motorcycle stunts (La Moto Ailée, Saut dans la Lune), but after a bad crash in Berlin 1904 she had to spend eight months recovering her health.

Acting career
Hélène Dutrieu had played a small role at the Déjazet theatre in 1903 (she played “Hélène” in “Môssieu le Maire“ by Gustave Stoskopf ). After regaining her health she went back to the theatre playing comedy in 1906–1909. She performed on stages such as Théâtre des Capucines, Théâtre des Mathurins, Porte Saint-Martin and Théâtre Michel.

Achievements in aviation

Dutrieu learned to fly using a Santos-Dumont Demoiselle monoplane in early 1910. On 19 April 1910 she reputedly became the first female pilot to fly with a passenger. On 25 November 1910 Dutrieu became the fourth woman in the world, and the first Belgian woman, licensed as an aeroplane pilot, receiving Aéro-Club de Belgique (Aero Club of Belgium) licence #27. Her appearances at air shows earned her the nickname the "Girl Hawk". There was a minor scandal early in her aviation career when it was revealed to the press that she did not wear a corset while flying.

She was also style conscious, wearing the first known high fashion pilot suit, designed by the Paris couturier Bernard In September 1910 Dutrieu flew non-stop from Ostend to Bruges, Belgium. From 26 September to 1 October she flew, frequently carrying passengers, at the aviation week in Burton-upon-Trent, England. She was the first woman pilot to stay airborne for more than an hour and on 21 December 1910 she became the first winner of the Coupe Femina (Femina Cup) for a non-stop flight of 167 km in 2 hours 35 minutes. In 1911 she regained the Coupe Femina temporarily with a flight of 254 km in 2 hours 58 minutes but that year's cup was eventually won by Marie Marvingt. In September 1911 Dutrieu travelled to the United States with her Farman III biplane. She competed for the women's altitude record and the Rodman-Wanamaker trophy, subsequently won by Matilde Moisant, at the Nassau Boulevard airfield meeting in Garden City, New York. In the same year Dutrieu beat 14 male pilots to win the Coppa del Re (King's Cup) in Florence, Italy. In 1912 she became the first woman to pilot a seaplane. Later the same year she won a prize in competition against four other seaplane pilots, including Réne Caudron, at Ouchy-Lausanne, Switzerland. In 1913 Dutrieu became the first woman aviator awarded membership of the Légion d'honneur (French Legion of Honour).

World War I and afterward
During World War I Dutrieu became an ambulance driver. Général Février put her in charge of the ambulances at Messimi Hospital. She later became the director of Campagne à Val-de Grâce military hospital. After the war she became a journalist. In 1922 she married Pierre Mortier and took French nationality. She later became vice president of the women's section of the Aéro-Club de France (Aero Club of France). In 1956 she created the Coupe Hélène Dutrieu-Mortier (Hélène Dutrieu-Mortier Cup) with a prize of 200,000 francs for the French or Belgian female pilot who made the longest non-stop flight each year.

Death
Hélène Dutrieu died in Paris, France, on 26 June 1961, at the age of 83.

Awards

 * 1898, awarded the Cross of St. Andre with diamonds by Leopold II of Belgium.
 * 1910 December 21, won the Aéro-Club de France's (Aero Club of France) Coupe Femina (Femina Cup).
 * 1913, named a member of the légion d'honneur (French Legion of Honour).

title: Hélène Dutrieu


text:
Marketing warfare strategies

Marketing warfare strategies represent a type of strategy, used in commerce and marketing, that tries to draw parallels between business and warfare and then applies the principles of military strategy to business situations, with competing firms considered as analogous to sides in a military conflict, and market share considered as analogous to territory in dispute. This view of marketing argues that in mature, low-growth markets, where real GDP growth is negative or low, commerce operates as a zero-sum game. One participant's gain is possible only at another participant's expense. Success depends on battling competitors for market share.

The use of marketing warfare strategies
Strategy is the organized deployment of resources to achieve specific objectives, something that business and warfare have in common. In the 1980s business strategists realized that there was a vast knowledge base stretching back thousands of years that they had barely examined. They turned to military strategy for guidance. Military strategy books like The Art of War by Sun Tzu, On War by von Clausewitz, and The Little Red Book by Mao Zedong became business classics.

From Sun Tzu they learned the tactical side of military strategy and specific tactical proscriptions. In regard to what business strategists call "first-mover advantage", Sun Tzu said: "Generally, he who occupies the field of battle first and awaits an enemy is at ease, he who comes later to the scene and rushes into the fight is weary." From Von Clausewitz they learned the dynamic and unpredictable nature of military strategy. Clausewitz felt that in a situation of chaos and confusion, strategy should be based on flexible principles. Strategy comes not from formulas or rules of engagement, but from adapting to what he called "friction" (minute-by-minute events). From Mao Zedong they learned the principles of guerrilla warfare.

The first major proponents of marketing warfare theories were Philip Kotler and J.B. Quinn. In an early description of business military strategy, Quinn claims that an effective strategy: "first probes and withdraws to determine opponents' strengths, forces opponents to stretch their commitments, then concentrates resources, attacks a clear exposure, overwhelms a selected market segment, builds a bridgehead in that market, and then regroups and expands from that base to dominate a wider field."

The main marketing warfare books were
 * Business War Games by Barrie James, 1985
 * Marketing Warfare by Al Ries and Jack Trout, 1986
 * Leadership Secrets of Attila the Hun by Wess Roberts, 1987

By the turn of the century, marketing warfare strategies had gone out of fashion. It was felt that they were limiting. There were many situations in which non-confrontational approaches were more appropriate. The Strategy of the Dolphin was developed in the mid-1990s to give guidance as to when to use aggressive strategies and when to use passive strategies. Today, most business strategists stress that considerable synergies and competitive advantages can be gained from collaboration, partnering, and co-operation. They stress how to divide up the market and how to grow the market. Such are the vicissitudes of business theories. Finally, a recent contribution to understanding and using marketing warfare strategies is the visual business war game proposed by S. Goria.

Marketing warfare strategies

 * Offensive marketing warfare strategies - They are used to secure competitive advantages; market leaders, runners-up or struggling competitors are usually attacked.
 * Defensive marketing warfare strategies - They are used to defend competitive advantages; lessen risk of being attacked, decrease effects of attacks, strengthen position.
 * Flanking marketing warfare strategies - They operate in areas of little importance to the competitor.
 * Guerrilla marketing warfare strategies - Attack, retreat, hide, then do it again, and again, until the competitor moves on to other markets.
 * Position defense - This is a strategy which utilizes its current position against the attacking opposition. In a business context, this is a strategy usually applied when a company has a dominant stake in the market place, usually a monopolized and controlled industry. Marketing with this type of strategy can be identified through barriers of entry. This is where a company has fortified its position by having key strongholds in the Marketing segment or brand identity or product familiarity. They may apply these areas through increasing the equity of the brand or repeat purchases other wise known as customer loyalty strategies (Shayne, Milligan. 2012) . E.g. Starbucks as a café giant promoted the free wifi connection to protect their market share against the competition that had first applied the concept (Jamie, Burns. 2013).
 * Mobile defense - By moving resources and creating new strategies and tactics, the intended goal is to create a moving target that is difficult to attack by the opposition. This also equips the defense to repel any attacks the opposition has in store. The interpretation in business explained by Shayne Milligan is when businesses introduce new products, replacement products, modifying existing products and repositioning products as well as changing the marketing segments, target markets or changing promotional focus. This type of defensive strategy is most likely incorporated by entrepreneurial companies with strong marketing research and marketing skills along with the ability to continuously develop their product line (Shayne Milligan. 2012).
 * Flanking position - Re-deploy your resources to discourage any type of flanking attack. This in business terms is developing new products in a marketing segment that you occupy. By expanding resources the business is able to strengthen their hold on the segment under threat (Shayne Milligan. 2012). E.g. Absolut Vodka had found a marketing segment that was leased served. In doing so they were able to capture this market by increasing prices by promoting premium vodka this tackled their competition Smirnoff in a space they did not allocate resources towards. This is known as Flanking marketing compared to the strategy of flanking position this was a successful attack towards the opposition (Jamie Burns. 2013).
 * Counter offense - This initially involves counterattacking the opposition that has attacked you. In business context this is where a counterattack is made on the oppositions weakest point (Shayne Milligan. 2012). Pre-emptive attack on any business must have some type of counter re-action this could be a move by the competition into its sales territory, price cutting, promotional blitz or product improvements. When faced with these competitive signs the options can be frontal deployment of resources by strategically developing new products or improving on products. The other option is finding the oppositions weakest point, which in military terms would be attacking the competitions main territory. E.g. Central DuPage, a suburban hospital located in Chicago, had been under invasion by competitors. Primary and urgent care centres had moved into the local suburban area and with the population rising it became and opportunity not just for the Central hospital DuPage but also for their competitors. In order for the local hospital to protect its share in the market it had to develop new physician offices located in under served areas. This was a counter offence by re-positioning themselves they were able to take in patients through their newly allocated offices where the physicians could refer their patients to the Central DuPage hospital (Naresh, K Malhotra. 1988).
 * Frontal attack - This strategy is specifically designed to engage the opposition with a head-on frontal assault. This also means using a substantial amount of resources and financial commitment when taking on a competitor with this strategy. From marketing to the production process, all elements are activated when initiating such an abrasive move. Advertising campaigns and new products are usually intensified to take on the competition where they are strongest; this is to weaken their market share and margins by cutting off their leading products and influencing their targeted audience to re-evaluate their loyalty to the brand or product. It is rare to encounter such strategies, as the process is quite expensive and time-consuming; this is also a high-risk venture if the competition has a strong counteroffensive attack, which can leave the attacking opponent open to counterstrikes. With resources already stretched this strategy is not for the faint-hearted. Shayne Milligan explains that this type of strategy is only used when the market space is homogenous, brand equity is low, customer loyalty is low, products are poorly differentiated, the competitor has relatively low resources or the attacker has stronger resources (Shayne Milligan. 2013). E.g. In 2011 the US shopping giant Target had entered the Canadian market with financial investments exceeding 4.4 billion dollars. Part of this initial investment was allocated towards purchasing 220 previously acquired stores of Zellers, a local Canadian merchandising mogul. With almost 10 million being spent on the refurbishing of each store as well as the hiring of 150–200 employees per store, the financial forecast for the Target Corporation was estimated to be around 6 billion per year by the year of 2017. This was not to be the case, as Target's initiating year in the Canadian retail market failed to achieve any realistic financial goals. Even though the success of Target's US stores were still in effect, the underestimated planning into the Canadian market was due to unforeseen economic variables as well as Canadian loyalty to stores also owned by US parent companies Wal-Mart and Costco, etc. Such competition had put the Target Corporation in a position at risk of making more losses than profits, which would also effect their US-based stores. Taking on Wal-Mart and other foreign and domestic competition has been unsuccessful, but Target continues to move towards dominating some part of the Canadian retail market (2015).
 * Envelopment strategy or encirclement strategy - This strategy is more broadly used as it focuses on subtle offensive attacks. In such cases the introduction of products that are similar to the competitor's products are developed to liberate the market share of the opposition's product line. When done properly this type of strategy can avoid a full-scale frontal assault. The objective is to find niches in the marketing space rather than the creating of products that directly compete against the competition. It is more of an indirect assault on the oppositions market share. Shayne Milligan suggests this strategy be used when the market is loosely segmented, some segments are free of larger competitors, the attacker has strong product development resources, the attacker has enough resources to operate in multiple segments simultaneously and the attacker has a decentralized organizational structure (Shayne Milligan. 2013). E.g. Republic Health Corporation, which is a chain of health care centers located in Dallas, developed an advertising campaign called “Step Lively”. This was specifically incorporated to focus on pricing, product form, sales promotion and advertising. The strategy was to create incentives through discounted foot examinations and free home meals after hospitalization as well as purchasing gift cards for a new shoe's. They focused on one area at a time against competitors while funneling specific treatments for patients into one hospital in each area at a time (Naresh K, Malhotra. 1988).
 * Leapfrog strategy - Bypassing is a less indirect strategy compared to the alternative options. In business terms this can be achieved through technological advancements or creating new segments that have not been developed as of yet (Naresh K, Malhotra. 1988).
 * Guerilla - Guerrilla strategies used usually consists of small incremental attacks by using unconventional methods against a larger opposition. In a business context this can most commonly be used by smaller firms on the bigger competition. These are tactically made forms of communications between the consumers influence of the bigger competitors specifically targeting a market segment that is heavily influenced by the competing opposition. They may be short burst attacks through price cuts, supply deterrence, executive raids or a promotional blitz, even legal actions against the competition or negative publicity. But this type of strategy must have some type of disengage tactic, as a full on confrontational assault could be disastrous for a small firm (Naresh K, Malhotra. 1988). E.g. In 2005 a telecommunications company based in Romania had launched itself in already emerging market of mobile phones. They did this by acquiring an already unsuccessful firm but re-branding the company and using the already established networks. Two rival competitors at that time had complete control of the telecommunications sector in Romania. In order for the smaller firm to compete, they had to resort to two marketing warfare tactics: flanking attacks as well as guerrilla strategies. They were able to penetrate two segments, pre-pay and post-pay. By not taking the competition head on through more heavily operated segments of the industry at that time, they were able to find an area that was less productive for the larger telecommunication providers. The small firm was able to increase 95% of their coverage to 82% of their networks distribution area making mobile phone more affordable and assessable to their customer base initially targeting a young audience. This was unexpected by the two larger telecommunications providers even though they still controlled a larger portion of the telecom industry the small firm was able to become a medium-sized competitor within a short period of time (B, G, Cernat. G, L, Constantin. A, Chiciudean. 2008).

Companies typically use many strategies concurrently, some defensive, some offensive, and always some deterrent. According to the business literature of the period, offensive strategies were more important than defensive one. Defensive strategies were used when needed, but an offensive strategy was requisite. Only by offensive strategies, were market gains made. DAt best, defensive strategies could save you from falling too far behind.

The marketing warfare literature also examined leadership and motivation, intelligence gathering, types of marketing weapons, logistics, and communications.

title: Marketing warfare strategies