text:
Connection (vector bundle)

In mathematics, and especially differential geometry and gauge theory, a connection on a fiber bundle is a device that defines a notion of parallel transport on the bundle; that is, a way to "connect" or identify fibers over nearby points. The most common case is that of a linear connection on a vector bundle, for which the notion of parallel transport must be linear. A linear connection is equivalently specified by a covariant derivative, an operator that differentiates sections of the bundle along tangent directions in the base manifold, in such a way that parallel sections have derivative zero. Linear connections generalize, to arbitrary vector bundles, the Levi-Civita connection on the tangent bundle of a pseudo-Riemannian manifold, which gives a standard way to differentiate vector fields. Nonlinear connections generalize this concept to bundles whose fibers are not necessarily linear.

Linear connections are also called Koszul connections after Jean-Louis Koszul, who gave an algebraic framework for describing them.

This article defines the connection on a vector bundle using a common mathematical notation which de-emphasizes coordinates. However, other notations are also regularly used: in general relativity, vector bundle computations are usually written using indexed tensors; in gauge theory, the endomorphisms of the vector space fibers are emphasized. The different notations are equivalent, as discussed in the article on metric connections (the comments made there apply to all vector bundles).

Motivation
Let $M$ be a differentiable manifold, such as Euclidean space. A vector-valued function $$M \to \mathbb{R}^n$$ can be viewed as a section of the trivial vector bundle $$M\times \mathbb{R}^n \to M.$$ One may consider a section of a general differentiable vector bundle, and it is therefore natural to ask if it is possible to differentiate a section, as a generalization of how one differentiates a function on $M$.

The model case is to differentiate a function $$X: \mathbb{R}^n \to \mathbb{R}^m$$ on Euclidean space $$\mathbb{R}^n$$. In this setting the derivative $$dX$$ at a point $$x\in \mathbb{R}^n$$ in the direction $$v\in \mathbb{R}^n$$ may be defined by the standard formula
 * $$dX(v)(x) = \lim_{t\to 0} \frac{X(x+tv) - X(x)}{t}.$$

For every $$x\in \mathbb{R}^n$$, this defines a new vector $$dX(v)(x)\in\mathbb{R}^m.$$

When passing to a section $$X$$ of a vector bundle $$E$$ over a manifold $$M$$, one encounters two key issues with this definition. Firstly, since the manifold has no linear structure, the term $$x+tv$$ makes no sense on $$M$$. Instead one takes a path $$\gamma: (-1,1) \to M$$ such that $$\gamma(0) = x, \gamma'(0) = v$$ and computes
 * $$dX(v)(x) = \lim_{t\to 0} \frac{X(\gamma(t)) - X(\gamma(0))}{t}.$$

However this still does not make sense, because $$X(\gamma(t))$$ and $$X(\gamma(0))$$ are elements of the distinct vector spaces $$E_{\gamma(t)}$$ and $$E_x.$$ This means that subtraction of these two terms is not naturally defined.

The problem is resolved by introducing the extra structure of a connection to the vector bundle. There are at least three perspectives from which connections can be understood. When formulated precisely, all three perspectives are equivalent.

 * 1) (Parallel transport) A connection can be viewed as assigning to every differentiable path $$\gamma$$ a linear isomorphism $$P_t^{\gamma} : E_{\gamma(t)} \to E_{x}$$ for all $$t.$$ Using this isomorphism one can transport $$X(\gamma(t))$$ to the fibre $$E_x$$ and then take the difference; explicitly, $$\nabla_vX = \lim_{t\to 0} \frac{P_{t}^{\gamma} X(\gamma(t)) - X(\gamma(0))}{t}.$$In order for this to depend only on $$v,$$ and not on the path $$\gamma$$ extending $$v,$$ it is necessary to place restrictions (in the definition) on the dependence of $$P_t^{\gamma}$$ on $$\gamma.$$ This is not straightforward to formulate, and so this notion of "parallel transport" is usually derived as a by-product of other ways of defining connections. In fact, the following notion of "Ehresmann connection" is nothing but an infinitesimal formulation of parallel transport.
 * 2) (Ehresmann connection) The section $$X$$ may be viewed as a smooth map from the smooth manifold $$M$$ to the smooth manifold $$E.$$ As such, one may consider the pushforward $$dX(v),$$ which is an element of the tangent space $$T_{X(x)}E.$$ In Ehresmann's formulation of a connection, one chooses a way of assigning, to each $$x$$ and every $$e\in E_x,$$ a direct sum decomposition of $$T_{X(x)}E$$ into two linear subspaces, one of which is the natural embedding of $$E_x.$$ With this additional data, one defines $$\nabla_vX$$ by projecting $$dX(v)$$ to be valued in $$E_x.$$ In order to respect the linear structure of a vector bundle, one imposes additional restrictions on how the direct sum decomposition of $$T_{e}E$$ moves as $e$ is varied over a fiber.
 * 3) (Covariant derivative) The standard derivative $$dX(v)$$ in Euclidean contexts satisfies certain dependencies on $$X$$ and $$v,$$ the most fundamental being linearity. A covariant derivative is defined to be any operation $$(v,X)\mapsto\nabla_vX$$ which mimics these properties, together with a form of the product rule.

Unless the base is zero-dimensional, there are always infinitely many connections which exist on a given differentiable vector bundle, and so there is always a corresponding choice of how to differentiate sections. Depending on context, there may be distinguished choices, for instance those which are determined by solving certain partial differential equations. In the case of the tangent bundle, any pseudo-Riemannian metric (and in particular any Riemannian metric) determines a canonical connection, called the Levi-Civita connection.

Formal definition
Let $$E\to M$$ be a smooth real vector bundle over a smooth manifold $$M$$. Denote the space of smooth sections of $$E\to M$$ by $$\Gamma(E)$$. A covariant derivative on $$E\to M$$ is either of the following equivalent structures: Beyond using the canonical identification between the vector space $$T_x^\ast M\otimes E_x$$ and the vector space of linear maps $$T_xM\to E_x,$$ these two definitions are identical and differ only in the language used.
 * 1) an $$\mathbb{R}$$-linear map $$\nabla : \Gamma(E) \to \Gamma(T^*M\otimes E)$$ such that the product rule $$\nabla(fs) = df\otimes s + f\nabla s$$ holds for all smooth functions $$f$$ on $$M$$ and all smooth sections $$s$$ of $$E.$$
 * 2) an assignment, to any smooth section $s$ and every $$x\in M$$, of a $$\mathbb{R}$$-linear map $$(\nabla s)_x:T_xM\to E_x$$ which depends smoothly on $x$ and such that $$\nabla(a_1s_1+a_2s_2)=a_1\nabla s_1+a_2\nabla s_2$$ for any two smooth sections $$s_1,s_2$$ and any real numbers $$a_1,a_2,$$ and such that for every smooth function $$f$$, $$\nabla(fs)$$ is related to $$\nabla s$$ by $$\big(\nabla(fs)\big)_x(v)=df(v)s(x)+f(x)(\nabla s)_x(v)$$ for any $$x\in M$$ and $$v\in T_xM.$$

It is typical to denote $$(\nabla s)_x(v)$$ by $$\nabla_vs,$$ with $$x$$ being implicit in $$v.$$ With this notation, the product rule in the second version of the definition given above is written
 * $$\nabla_v(fs)=df(v)s+f\nabla_vs.$$

Remark. In the case of a complex vector bundle, the above definition is still meaningful, but is usually taken to be modified by changing "real" and "$$\mathbb{R}$$" everywhere they appear to "complex" and "$$\mathbb{C}.$$" This places extra restrictions, as not every real-linear map between complex vector spaces is complex-linear. There is some ambiguity in this distinction, as a complex vector bundle can also be regarded as a real vector bundle.

Induced connections
Given a vector bundle $$E\to M$$, there are many associated bundles to $$E$$ which may be constructed, for example the dual vector bundle $$E^*$$, tensor powers $$E^{\otimes k}$$, symmetric and antisymmetric tensor powers $$S^k E, \Lambda^k E$$, and the direct sums $$E^{\oplus k}$$. A connection on $$E$$ induces a connection on any one of these associated bundles. The ease of passing between connections on associated bundles is more elegantly captured by the theory of principal bundle connections, but here we present some of the basic induced connections.

Dual connection
Given $$\nabla$$ a connection on $$E$$, the induced dual connection $$\nabla^*$$ on $$E^*$$ is defined implicitly by
 * $$ d(\langle \xi, s \rangle)(X) = \langle \nabla_X^* \xi, s \rangle + \langle \xi, \nabla_X s \rangle.$$

Here $$X\in \Gamma(TM)$$ is a smooth vector field, $$s\in \Gamma(E)$$ is a section of $$E$$, and $$\xi \in \Gamma(E^*)$$ a section of the dual bundle, and $$\langle \cdot, \cdot \rangle$$ the natural pairing between a vector space and its dual (occurring on each fibre between $$E$$ and $$E^*$$), i.e., $$\langle \xi , s \rangle:= \xi(s)$$. Notice that this definition is essentially enforcing that $$\nabla^*$$ be the connection on $$E^*$$ so that a natural product rule is satisfied for pairing $$ \langle \cdot, \cdot \rangle$$.

Tensor product connection
Given $$\nabla^E, \nabla^F$$ connections on two vector bundles $$E, F\to M$$, define the tensor product connection by the formula
 * $$ (\nabla^E \otimes \nabla^F)_X(s\otimes t) = \nabla_X^E (s) \otimes t + s\otimes \nabla_X^F (t). $$

Here we have $$s\in \Gamma(E), t\in \Gamma(F), X\in \Gamma(TM)$$. Notice again this is the natural way of combining $$\nabla^E, \nabla^F$$ to enforce the product rule for the tensor product connection. By repeated application of the above construction applied to the tensor product $$E^{\otimes k} = (E^{\otimes (k-1)}) \otimes E$$, one also obtains the tensor power connection on $$E^{\otimes k}$$ for any $$k\ge 1$$ and vector bundle $$E$$.

Direct sum connection
The direct sum connection is defined by

 * $$ (\nabla^E \oplus \nabla^F)_X (s\oplus t) = \nabla_X^E (s) \oplus \nabla_X^F (t),$$

where $$s\oplus t\in \Gamma(E\oplus F)$$.

Symmetric and exterior power connections
Since the symmetric power and exterior power of a vector bundle may be viewed naturally as subspaces of the tensor power, $$S^k E, \Lambda^k E \subset E^{\otimes k}$$, the definition of the tensor product connection applies in a straightforward manner to this setting. Indeed, since the symmetric and exterior algebras sit inside the tensor algebra as direct summands, and the connection $$\nabla$$ respects this natural splitting, one can simply restrict $$\nabla$$ to these summands. Explicitly, define the symmetric product connection by
 * $$\nabla^{\odot 2}_X(s\cdot t) = \nabla_X s \odot t + s \odot \nabla_X t$$

and the exterior product connection by
 * $$\nabla^{\wedge 2}_X (s\wedge t) = \nabla_X s \wedge t + s\wedge \nabla_X t$$

for all $$s,t\in \Gamma(E), X\in \Gamma(TM)$$. Repeated applications of these products gives induced symmetric power and exterior power connections on $$S^k E$$ and $$\Lambda^k E$$ respectively.

Endomorphism connection
Finally, one may define the induced connection $$\nabla^{\operatorname{End}{E}}$$ on the vector bundle of endomorphisms $$\operatorname{End}(E) = E^* \otimes E$$, the endomorphism connection. This is simply the tensor product connection of the dual connection $$\nabla^*$$ on $$E^*$$ and $$\nabla$$ on $$E$$. If $$s\in \Gamma(E)$$ and $$u\in \Gamma(\operatorname{End}(E))$$, so that the composition $$u(s) \in \Gamma(E)$$ also, then the following product rule holds for the endomorphism connection:
 * $$\nabla_X(u(s)) = \nabla_X^{\operatorname{End}(E)} (u) (s) + u(\nabla_X (s)).$$

By reversing this equation, it is possible to define the endomorphism connection as the unique connection satisfying

 * $$ \nabla_X^{\operatorname{End}(E)} (u) (s) = \nabla_X(u(s)) - u(\nabla_X(s))$$

for any $$u,s,X$$, thus avoiding the need to first define the dual connection and tensor product connection.

Any associated bundle
Given a vector bundle $$E$$ of rank $$r$$, and any representation $$\rho: \mathrm{GL}(r,\mathbb{K}) \to G$$ into a linear group $$G\subset \mathrm{GL}(V)$$, there is an induced connection on the associated vector bundle $$F = E\times_\rho V$$. This theory is most succinctly captured by passing to the principal bundle connection on the frame bundle of $$E$$ and using the theory of principal bundles. Each of the above examples can be seen as special cases of this construction: the dual bundle corresponds to the inverse transpose (or inverse adjoint) representation, the tensor product to the tensor product representation, the direct sum to the direct sum representation, and so on.

Exterior covariant derivative and vector-valued forms
Let $$E\to M$$ be a vector bundle. An $E$-valued differential form of degree $$r$$ is a section of the tensor product bundle:

 * $$\bigwedge^rT^*M \otimes E.$$

The space of such forms is denoted by

 * $$\Omega^r(E) = \Omega^r(M;E) = \Gamma \left (\bigwedge^rT^*M \otimes E \right ) = \Omega^r(M) \otimes_{C^{\infty}(M)} \Gamma(E),$$

where the last tensor product denotes the tensor product of modules over the ring of smooth functions on $$M$$.

An $$E$$-valued 0-form is just a section of the bundle $$E$$. That is,

 * $$\Omega^0(E) = \Gamma(E).$$

In this notation a connection on $$E\to M$$ is a linear map

 * $$\nabla:\Omega^0(E) \to \Omega^1(E).$$

A connection may then be viewed as a generalization of the exterior derivative to vector bundle valued forms. In fact, given a connection $$\nabla$$ on $$E$$ there is a unique way to extend $$\nabla$$ to an exterior covariant derivative

 * $$d_{\nabla}: \Omega^r(E) \to \Omega^{r+1}(E).$$

This exterior covariant derivative is defined by the following Leibniz rule, which is specified on simple tensors of the form $$\omega\otimes s$$ and extended linearly:

 * $$d_\nabla (\omega \otimes s) = d\omega \otimes s + (-1)^{\deg \omega} \omega \wedge \nabla s$$

where $$\omega \in \Omega^r(M)$$ so that $$\deg \omega = r$$, $$s\in \Gamma(E)$$ is a section, and $$\omega \wedge \nabla s$$ denotes the $$(r+1)$$-form with values in $$E$$ defined by wedging $$\omega$$ with the one-form part of $$\nabla s$$. Notice that for $$E$$-valued 0-forms, this recovers the normal Leibniz rule for the connection $$\nabla$$.

Unlike the ordinary exterior derivative, one generally has $$d_{\nabla}^2 \ne 0$$. In fact, $$d_{\nabla}^2$$ is directly related to the curvature of the connection $$\nabla$$ (see below).

Affine properties of the set of connections
Every vector bundle over a manifold admits a connection, which can be proved using partitions of unity. However, connections are not unique. If $$\nabla_1$$ and $$\nabla_2$$ are two connections on $$E\to M$$ then their difference is a $C^{\infty}(M)$-linear operator. That is,
 * $$(\nabla_1 - \nabla_2)(fs) = f(\nabla_1s - \nabla_2s)$$

for all smooth functions $$f$$ on $$M$$ and all smooth sections $$s$$ of $$E$$. It follows that the difference $$\nabla_1-\nabla_2$$ can be uniquely identified with a one-form on $$M$$ with values in the endomorphism bundle $$\operatorname{End}(E) = E^* \otimes E$$:
 * $$\nabla_1 - \nabla_2 \in \Omega^1(M; \mathrm{End}\,E).$$

Conversely, if $$\nabla$$ is a connection on $$E$$ and $$A$$ is a one-form on $$M$$ with values in $$\operatorname{End}(E)$$, then $$\nabla + A$$ is a connection on $$E$$.

In other words, the space of connections on $$E$$ is an affine space for $$\Omega^1(\operatorname{End}(E))$$. This affine space is commonly denoted $$\mathcal{A}$$.

Relation to principal and Ehresmann connections
Let $$E\to M$$ be a vector bundle of rank $$k$$ and let $$\mathcal{F}(E)$$ be the frame bundle of $$E$$. Then a (principal) connection on $$\mathcal{F}(E)$$ induces a connection on $$E$$. First note that sections of $$E$$ are in one-to-one correspondence with right-equivariant maps $$\mathcal{F}(E)\to \mathbb{R}^k$$. (This can be seen by considering the pullback of $$E$$ over $$\mathcal{F}(E)\to M$$, which is isomorphic to the trivial bundle $$\mathcal{F}(E)\times \mathbb{R}^k$$.) Given a section $$s$$ of $$E$$ let the corresponding equivariant map be $$\psi(s)$$. The covariant derivative on $$E$$ is then given by
 * $$\psi(\nabla_Xs) = X^H(\psi(s))$$

where $$X^H$$ is the horizontal lift of $$X$$ from $$M$$ to $$\mathcal{F}(E)$$. (Recall that the horizontal lift is determined by the connection on $$\mathcal{F}(E)$$.)

Conversely, a connection on $$E$$ determines a connection on $$\mathcal{F}(E)$$, and these two constructions are mutually inverse.

A connection on $$E$$ is also determined equivalently by a linear Ehresmann connection on $$E$$. This provides one method to construct the associated principal connection.

The induced connections discussed in can be constructed as connections on other associated bundles to the frame bundle of $$E$$, using representations other than the standard representation used above. For example if $$\rho$$ denotes the standard representation of $$\operatorname{GL}(k,\mathbb{R})$$ on $$\mathbb{R}^k$$, then the associated bundle to the representation $$\rho \oplus \rho$$ of $$\operatorname{GL}(k,\mathbb{R})$$ on $$\mathbb{R}^k \oplus \mathbb{R}^k$$ is the direct sum bundle $$E\oplus E$$, and the induced connection is precisely that which was described above.

Local expression
Let $$E\to M$$ be a vector bundle of rank $$k$$, and let $$U$$ be an open subset of $$M$$ over which $$E$$ trivialises. Therefore over the set $$U$$, $$E$$ admits a local smooth frame of sections

 * $$\mathbf{e} = (e_1, \dots, e_k);\quad e_i: U \to \left.E\right|_U.$$

Since the frame $$\mathbf{e}$$ defines a basis of the fibre $$E_x$$ for any $$x\in U$$, one can expand any local section $$s:U\to \left.E\right|_U$$ in the frame as
 * $$s = \sum_{i=1}^k s^i e_i$$

for a collection of smooth functions $$s^1, \dots, s^k: U \to \mathbb{R}$$.

Given a connection $$\nabla$$ on $$E$$, it is possible to express $$\nabla$$ over $$U$$ in terms of the local frame of sections, by using the characteristic product rule for the connection. For any basis section $$e_i$$, the quantity $$\nabla(e_i)\in \Omega^1(U) \otimes \Gamma(U,E)$$ may be expanded in the local frame $$\mathbf{e}$$ as

 * $$ \nabla (e_i) = \sum_{j=1}^k A_i^{\ j} \otimes e_j,$$

where $$A_i^{\ j}\in \Omega^1(U);\, j=1,\dots,k$$ are a collection of local one-forms. These forms can be put into a matrix of one-forms defined by

 * $$ A = \begin{pmatrix} A_1^{\ 1} & \cdots & A_k^{\ 1} \\ \vdots & \ddots & \vdots \\ A_1^{\ k} & \cdots & A_k^{\ k} \end{pmatrix}\in \Omega^1(U, \operatorname{End}(\left.E\right|_U))$$

called the local connection form of $$\nabla$$ over $$U$$. The action of $$\nabla$$ on any section $$s: U \to \left.E\right|_U$$ can be computed in terms of $$A$$ using the product rule as

 * $$\nabla(s) = \sum_{j=1}^k \left(ds^j + \sum_{i=1}^k A_i^{\ j} s^i\right) \otimes e_j.$$

If the local section $$s$$ is also written in matrix notation as a column vector using the local frame $$\mathbf{e}$$ as a basis,

 * $$ s = \begin{pmatrix} s^1 \\ \vdots \\ s^k\end{pmatrix},$$

then using regular matrix multiplication one can write
 * $$\nabla(s) = ds + As$$

where $$ds$$ is shorthand for applying the exterior derivative $$d$$ to each component of $$s$$ as a column vector. In this notation, one often writes locally that $$\left.\nabla\right|_U = d+A$$. In this sense a connection is locally completely specified by its connection one-form in some trivialisation.

As explained in, any connection differs from another by an endomorphism-valued one-form. From this perspective, the connection one-form $$A$$ is precisely the endomorphism-valued one-form such that the connection $$\left.\nabla\right|_U$$ on $$\left.E\right|_U$$ differs from the trivial connection $$d$$ on $$\left.E\right|_U$$, which exists because $$U$$ is a trivialising set for $$E$$.

Relationship to Christoffel symbols
In pseudo-Riemannian geometry, the Levi-Civita connection is often written in terms of the Christoffel symbols $$\Gamma_{ij}^{\ \ k}$$ instead of the connection one-form $$A$$. It is possible to define Christoffel symbols for a connection on any vector bundle, and not just the tangent bundle of a pseudo-Riemannian manifold. To do this, suppose that in addition to $$U$$ being a trivialising open subset for the vector bundle $$E\to M$$, that $$U$$ is also a local chart for the manifold $$M$$, admitting local coordinates $$\mathbf{x} = (x^1,\dots,x^n);\quad x^i: U \to \mathbb{R}$$.

In such a local chart, there is a distinguished local frame for the differential one-forms given by $$(dx^1,\dots,dx^n)$$, and the local connection one-forms $$A_i^{ j}$$ can be expanded in this basis as

 * $$A_i^{\ j} = \sum_{\ell=1}^n \Gamma_{\ell i}^{\ \ j} dx^\ell$$

for a collection of local smooth functions $$\Gamma_{\ell i}^{\ \ j} : U \to \mathbb{R}$$, called the Christoffel symbols of $$\nabla$$ over $$U$$. In the case where $$E=TM$$ and $$\nabla$$ is the Levi-Civita connection, these symbols agree precisely with the Christoffel symbols from pseudo-Riemannian geometry.

The expression for how $$\nabla$$ acts in local coordinates can be further expanded in terms of the local chart $$U$$ and the Christoffel symbols, to be given by

 * $$ \nabla(s) = \sum_{i,j=1}^k \sum_{\ell=1}^n \left(\frac{\partial s^j}{\partial x^\ell} + \Gamma_{\ell i}^{\ \ j} s^i\right) dx^\ell \otimes e_j.$$

Contracting this expression with the local coordinate tangent vector $$\frac{\partial}{\partial x^\ell}$$ leads to
 * $$ \nabla_{\frac{\partial}{\partial x^\ell}} (s) = \sum_{i,j=1}^k \left(\frac{\partial s^j}{\partial x^\ell} + \Gamma_{\ell i}^{\ \ j} s^i\right) e_j.$$

This defines a collection of $$n$$ locally defined operators

 * $$\nabla_\ell: \Gamma(U,E) \to \Gamma(U,E);\quad \nabla_\ell(s) := \sum_{i,j=1}^k \left(\frac{\partial s^j}{\partial x^\ell} + \Gamma_{\ell i}^{\ \ j} s^i\right)e_j,$$

with the property that

 * $$\nabla(s) = \sum_{\ell=1}^n dx^\ell \otimes \nabla_\ell(s).$$

Change of local trivialisation
Suppose $$\mathbf{e'}$$ is another choice of local frame over the same trivialising set $$U$$, so that there is a matrix $$g=(g_i^{\ j})$$ of smooth functions relating $$\mathbf{e}$$ and $$\mathbf{e'}$$, defined by
 * $$e_i = \sum_{j=1}^k g_i^{\ j} e'_j.$$

Tracing through the construction of the local connection form $$A$$ for the frame $$\mathbf{e}$$, one finds that the connection one-form $$A'$$ for $$\mathbf{e'}$$ is given by
 * $${A'}_i^{\ j} = \sum_{p,q=1}^k g_p^{\ j} A_q^{\ p} {(g^{-1})}_i^{\ q} - \sum_{p=1}^k (dg)_p^{\ j} {(g^{-1})}_i^{\ p}$$

where $$g^{-1} = \left({(g^{-1})}_i^{\ j}\right)$$ denotes the inverse matrix to $$g$$. In matrix notation this may be written
 * $$A' = g A g^{-1} - (dg)g^{-1}$$

where $$dg$$ is the matrix of one-forms given by taking the exterior derivative of the matrix $$g$$ component-by-component.

In the case where $$E=TM$$ is the tangent bundle and $$g$$ is the Jacobian of a coordinate transformation of $$M$$, the lengthy formulae for the transformation of the Christoffel symbols of the Levi-Civita connection can be recovered from the more succinct transformation laws of the connection form above.

Parallel transport and holonomy
A connection $$\nabla$$ on a vector bundle $$E\to M$$ defines a notion of parallel transport on $$E$$ along a curve in $$M$$. Let $$\gamma: [0,1]\to M$$ be a smooth path in $$M$$. A section $$s$$ of $$E$$ along $$\gamma$$ is said to be parallel if
 * $$\nabla_{\dot\gamma(t)}s = 0$$

for all $$t\in [0,1]$$. Equivalently, one can consider the pullback bundle $$\gamma^* E$$ of $$E$$ by $$\gamma$$. This is a vector bundle over $$[0,1]$$ with fiber $$E_{\gamma(t)}$$ over $$t\in [0,1]$$. The connection $$\nabla$$ on $$E$$ pulls back to a connection on $$\gamma^* E$$. A section $$s$$ of $$\gamma^* E$$ is parallel if and only if $$\gamma^* \nabla(s) = 0$$.

Suppose $$\gamma$$ is a path from $$x$$ to $$y$$ in $$M$$. The above equation defining parallel sections is a first-order ordinary differential equation (cf. local expression above) and so has a unique solution for each possible initial condition. That is, for each vector $$v$$ in $$E_x$$ there exists a unique parallel section $$s$$ of $$\gamma^* E$$ with $$s(0) = v$$. Define a parallel transport map
 * $$\tau_\gamma : E_x \to E_y\,$$

by $$\tau_\gamma(v) = s(1)$$. It can be shown that $$\tau_\gamma$$ is a linear isomorphism, with inverse given by following the same procedure with the reversed path $$\gamma^-$$ from $$y$$ to $$x$$.

Parallel transport can be used to define the holonomy group of the connection $$\nabla$$ based at a point $$x$$ in $$M$$. This is the subgroup of $$\operatorname{GL}(E_x)$$ consisting of all parallel transport maps coming from loops based at $$x$$:
 * $$\mathrm{Hol}_x = \{\tau_\gamma : \gamma \text{ is a loop based at } x\}.\,$$

The holonomy group of a connection is intimately related to the curvature of the connection.

The connection can be recovered from its parallel transport operators as follows. If $$X\in \Gamma(TM)$$ is a vector field and $$s\in \Gamma(E)$$ a section, at a point $$x\in M$$ pick an integral curve $$\gamma: (-\varepsilon, \varepsilon) \to M$$ for $$X$$ at $$x$$. For each $$t\in (-\varepsilon, \varepsilon)$$ we will write $$\tau_t : E_{\gamma(t)} \to E_x$$ for the parallel transport map traveling along $$\gamma$$ from $$t$$ to $$0$$. In particular for every $$t\in (-\varepsilon, \varepsilon)$$, we have $$\tau_t s(\gamma(t)) \in E_x$$. Then $$t\mapsto \tau_t s(\gamma(t))$$ defines a curve in the vector space $$E_x$$, which may be differentiated. The covariant derivative is recovered as
 * $$\nabla_X s(x) = \frac{d}{dt} \left( \tau_t s(\gamma(t)) \right)_{t=0}.$$

This demonstrates that an equivalent definition of a connection is given by specifying all the parallel transport isomorphisms $$\tau_{\gamma}$$ between fibres of $$E$$ and taking the above expression as the definition of $$\nabla$$.

Curvature
The curvature of a connection $$\nabla$$ on $$E\to M$$ is a 2-form $$F_{\nabla}$$ on $$M$$ with values in the endomorphism bundle $$\operatorname{End}(E) = E^* \otimes E$$. That is,
 * $$F_\nabla \in \Omega^2(\mathrm{End}(E)) = \Gamma(\Lambda^2T^*M \otimes \mathrm{End}(E)).$$

It is defined by the expression
 * $$F_\nabla(X,Y)(s) = \nabla_X\nabla_Y s- \nabla_Y\nabla_X s- \nabla_{[X,Y]}s$$

where $$X$$ and $$Y$$ are tangent vector fields on $$M$$ and $$s$$ is a section of $$E$$. One must check that $$F_{\nabla}$$ is $C^{\infty}(M)$-linear in both $$X$$ and $$Y$$ and that it does in fact define a bundle endomorphism of $$E$$.

As mentioned above, the covariant exterior derivative $$d_{\nabla}$$ need not square to zero when acting on $$E$$-valued forms. The operator $$d_{\nabla}^2$$ is, however, strictly tensorial (i.e. $$C^{\infty}(M)$$-linear). This implies that it is induced from a 2-form with values in $$\operatorname{End}(E)$$. This 2-form is precisely the curvature form given above. For an $$E$$-valued form $$\sigma$$ we have
 * $$(d_\nabla)^2\sigma = F_\nabla\wedge\sigma.$$

A flat connection is one whose curvature form vanishes identically.

Local form and Cartan's structure equation
The curvature form has a local description called Cartan's structure equation. If $$\nabla$$ has local form $$A$$ on some trivialising open subset $$U\subset M$$ for $$E$$, then
 * $$F_{\nabla} = dA + A \wedge A$$

on $$U$$. To clarify this notation, notice that $$A$$ is a endomorphism-valued one-form, and so in local coordinates takes the form of a matrix of one-forms. The operation $$d$$ applies the exterior derivative component-wise to this matrix, and $$A\wedge A$$ denotes matrix multiplication, where the components are wedged rather than multiplied.

In local coordinates $$\mathbf{x} = (x^1,\dots,x^n)$$ on $$M$$ over $$U$$, if the connection form is written $$A=A_\ell dx^\ell = (\Gamma_{\ell i}^{\ \ j}) dx^\ell$$ for a collection of local endomorphisms $$A_\ell = (\Gamma_{\ell i}^{\ \ j})$$, then one has

 * $$F_{\nabla} = \sum_{p,q=1}^n \frac{1}{2} \left( \frac{\partial A_q}{\partial x^p} - \frac{\partial A_p}{\partial x^q} + [A_p, A_q]\right) dx^p \wedge dx^q.$$

Further expanding this in terms of the Christoffel symbols $$\Gamma_{\ell i}^{\ \ j}$$ produces the familiar expression from Riemannian geometry. Namely if $$s=s^i e_i$$ is a section of $$E$$ over $$U$$, then

 * $$F_{\nabla}(s) = \sum_{i,j=1}^k \sum_{p,q=1}^n \frac{1}{2} \left( \frac{\partial \Gamma_{qi}^{\ \ j}}{\partial x^p} - \frac{\partial \Gamma_{pi}^{\ \ j}}{\partial x^q} + \Gamma_{pr}^{\ \ j} \Gamma_{qi}^{\ \ r} - \Gamma_{qr}^{\ \ j} \Gamma_{pi}^{\ \ r} \right) s^i dx^p \wedge dx^q \otimes e_j = \sum_{i,j=1}^k \sum_{p,q=1}^n R_{pqi}^{\ \ \ j} s^i dx^p\wedge dx^q \otimes e_j.$$

Here $$R=(R_{pqi}^{\ \ \ j})$$ is the full curvature tensor of $$F_{\nabla}$$, and in Riemannian geometry would be identified with the Riemannian curvature tensor.

It can be checked that if we define $$[A, A]$$ to be wedge product of forms but commutator of endomorphisms as opposed to composition, then $$A \wedge A = \frac{1}{2} [A, A]$$, and with this alternate notation the Cartan structure equation takes the form
 * $$F_{\nabla} = dA + \frac{1}{2} [A, A].$$

This alternate notation is commonly used in the theory of principal bundle connections, where instead we use a connection form $$\omega$$, a Lie algebra-valued one-form, for which there is no notion of composition (unlike in the case of endomorphisms), but there is a notion of a Lie bracket.

In some references (see for example ) the Cartan structure equation may be written with a minus sign:
 * $$F_{\nabla} = dA - A \wedge A.$$

This different convention uses an order of matrix multiplication that is different from the standard Einstein notation in the wedge product of matrix-valued one-forms.

Bianchi identity
A version of the second (differential) Bianchi identity from Riemannian geometry holds for a connection on any vector bundle. Recall that a connection $$\nabla$$ on a vector bundle $$E\to M$$ induces an endomorphism connection on $$\operatorname{End}(E)$$. This endomorphism connection has itself an exterior covariant derivative, which we ambiguously call $$d_{\nabla}$$. Since the curvature is a globally defined $$\operatorname{End}(E)$$-valued two-form, we may apply the exterior covariant derivative to it. The Bianchi identity says that
 * $$d_{\nabla} F_{\nabla} = 0$$.

This succinctly captures the complicated tensor formulae of the Bianchi identity in the case of Riemannian manifolds, and one may translate from this equation to the standard Bianchi identities by expanding the connection and curvature in local coordinates.

There is no analogue in general of the first (algebraic) Bianchi identity for a general connection, as this exploits the special symmetries of the Levi-Civita connection. Namely, one exploits that the vector bundle indices of $$E=TM$$ in the curvature tensor $$R$$ may be swapped with the cotangent bundle indices coming from $$T^*M$$ after using the metric to lower or raise indices. For example this allows the torsion-freeness condition $$\Gamma_{\ell i}^{\ \ j} = \Gamma_{i \ell}^{\ \ j}$$ to be defined for the Levi-Civita connection, but for a general vector bundle the $$\ell$$-index refers to the local coordinate basis of $$T^*M$$, and the $$i,j$$-indices to the local coordinate frame of $$E$$ and $$E^*$$ coming from the splitting $$\mathrm{End}(E)=E^* \otimes E$$. However in special circumstance, for example when the rank of $$E$$ equals the dimension of $$M$$ and a solder form has been chosen, one can use the soldering to interchange the indices and define a notion of torsion for affine connections which are not the Levi-Civita connection.

Gauge transformations
Given two connections $$\nabla_1, \nabla_2$$ on a vector bundle $$E\to M$$, it is natural to ask when they might be considered equivalent. There is a well-defined notion of an automorphism of a vector bundle $$E\to M$$. A section $$u\in \Gamma(\operatorname{End}(E))$$ is an automorphism if $$u(x)\in \operatorname{End}(E_x)$$ is invertible at every point $$x\in M$$. Such an automorphism is called a gauge transformation of $$E$$, and the group of all automorphisms is called the gauge group, often denoted $$\mathcal{G}$$ or $$\operatorname{Aut}(E)$$. The group of gauge transformations may be neatly characterised as the space of sections of the capital A adjoint bundle $$\operatorname{Ad}(\mathcal{F}(E))$$ of the frame bundle of the vector bundle $$E$$. This is not to be confused with the lowercase a adjoint bundle $$\operatorname{ad}(\mathcal{F}(E))$$, which is naturally identified with $$\operatorname{End}(E)$$ itself. The bundle $$\operatorname{Ad} \mathcal{F}(E) $$ is the associated bundle to the principal frame bundle by the conjugation representation of $$G=\operatorname{GL}(r)$$ on itself, $$g\mapsto ghg^{-1}$$, and has fibre the same general linear group $$\operatorname{GL}(r)$$ where $$\operatorname{rank} (E) = r$$. Notice that despite having the same fibre as the frame bundle $$\mathcal{F}(E)$$ and being associated to it, $$\operatorname{Ad}(\mathcal{F}(E))$$ is not equal to the frame bundle, nor even a principal bundle itself. The gauge group may be equivalently characterised as $$\mathcal{G} = \Gamma(\operatorname{Ad} \mathcal{F}(E)).$$

A gauge transformation $$u$$ of $$E$$ acts on sections $$s\in \Gamma(E)$$, and therefore acts on connections by conjugation. Explicitly, if $$\nabla$$ is a connection on $$E$$, then one defines $$u\cdot \nabla$$ by
 * $$(u\cdot \nabla)_X(s) = u(\nabla_X (u^{-1}(s))$$

for $$s\in \Gamma(E), X\in \Gamma(TM)$$. To check that $$u\cdot \nabla$$ is a connection, one verifies the product rule
 * $$ \begin{align} u\cdot \nabla(fs) &= u(\nabla(u^{-1}(fs)))\\&=u(\nabla(fu^{-1}(s)))\\&=u(df \otimes u^{-1}(s)) + u(f\nabla(u^{-1}(s)))\\&=df \otimes s + f u\cdot \nabla(s).\end{align}$$

It may be checked that this defines a left group action of $$\mathcal{G}$$ on the affine space of all connections $$\mathcal{A}$$.

Since $$\mathcal{A}$$ is an affine space modelled on $$\Omega^1(M, \operatorname{End}(E))$$, there should exist some endomorphism-valued one-form $$A_u\in \Omega^1(M, \operatorname{End}(E))$$ such that $$u\cdot \nabla = \nabla + A_u$$. Using the definition of the endomorphism connection $$\nabla^{\operatorname{End}(E)}$$ induced by $$\nabla$$, it can be seen that
 * $$u\cdot \nabla = \nabla - d^{\nabla}(u) u^{-1}$$

which is to say that $$A_u = - d^{\nabla}(u) u^{-1}$$.

Two connections are said to be gauge equivalent if they differ by the action of the gauge group, and the quotient space $$\mathcal{B} = \mathcal{A}/\mathcal{G}$$ is the moduli space of all connections on $$E$$. In general this topological space is neither a smooth manifold or even a Hausdorff space, but contains inside it the moduli space of Yang–Mills connections on $$E$$, which is of significant interest in gauge theory and physics.

Examples

 * A classical covariant derivative or affine connection defines a connection on the tangent bundle of M, or more generally on any tensor bundle formed by taking tensor products of the tangent bundle with itself and its dual.
 * A connection on $$\pi: \R^2\times \R \to \R$$ can be described explicitly as the operator
 * $$\nabla = d + \begin{bmatrix} f_{11}(x) & f_{12}(x) \\ f_{21}(x) & f_{22}(x) \end{bmatrix}dx$$
 * where $$d$$ is the exterior derivative evaluated on vector-valued smooth functions and $$f_{ij}(x)$$ are smooth. A section $$a \in \Gamma(\pi)$$ may be identified with a map
 * $$\begin{cases} \R \to \R^2 \\ x \mapsto (a_1(x),a_2(x)) \end{cases}$$
 * and then
 * $$\nabla(a)= \nabla \begin{bmatrix} a_1(x) \\ a_2(x) \end{bmatrix} = \begin{bmatrix} \frac{da_1(x)}{dx} + f_{11}(x)a_1(x) + f_{12}(x)a_2(x) \\ \frac{da_2(x)}{dx} + f_{21}(x)a_1(x) + f_{22}(x)a_2(x)\end{bmatrix}dx$$

 * If the bundle is endowed with a bundle metric, an inner product on its vector space fibers, a metric connection is defined as a connection that is compatible with the bundle metric.
 * A Yang-Mills connection is a special metric connection which satisfies the Yang-Mills equations of motion.
 * A Riemannian connection is a metric connection on the tangent bundle of a Riemannian manifold.
 * A Levi-Civita connection is a special Riemannian connection: the metric-compatible connection on the tangent bundle that is also torsion-free. It is unique, in the sense that given any Riemannian connection, one can always find one and only one equivalent connection that is torsion-free. "Equivalent" means it is compatible with the same metric, although the curvature tensors may be different; see teleparallelism. The difference between a Riemannian connection and the corresponding Levi-Civita connection is given by the contorsion tensor.
 * The exterior derivative is a flat connection on $$E=M \times \R$$ (the trivial line bundle over M).
 * More generally, there is a canonical flat connection on any flat vector bundle (i.e. a vector bundle whose transition functions are all constant) which is given by the exterior derivative in any trivialization.

title: Connection (vector bundle)


text:
Nihon University

Nihon University (日本大学), abbreviated as Nichidai (日大), is a private research university in Japan. Its predecessor, Nihon Law School (currently the Department of Law), was founded by Yamada Akiyoshi, the Minister of Justice, in 1889. It is one of Japan's leading private universities. The university's name is derived from the Japanese word "Nihon" meaning Japan. Nihon University now has "16 colleges and 87 departments, 20 postgraduate schools, 1 junior college which is composed of 5 departments, 1 correspondence division, 32 research institutes and 3 hospitals." The number of students exceeds 70,000 and is the largest in Japan.

University profile
Most of the university's campuses are in the Kantō region, with the vast majority in Tokyo or surrounding areas, although two campuses are as far away from Tokyo as Shizuoka Prefecture and Fukushima Prefecture. These campuses mostly accommodate single colleges or schools (gakubu (学部) in Japanese). In December 2016 the university acquired the former Newcastle Court House in, New South Wales, Australia for A$6.6 million as its inaugural international campus.

The university comprises a federation of colleges and institutes known for having produced numerous Japanese CEOs. Its College of Art (日芸 — Nichigei), located right next to Ekoda train station in Tokyo's Nerima ward, is well-known for producing many artists who represent Japan in photography, theater, and cinema. In addition, the university has over 20 affiliated high schools bearing its name across Japan, from which a significant number of students go on to study at the institution as undergraduates.

Colleges and departments

 * College of Law (1889-; Chiyoda, Tokyo & Saitama, Saitama)
 * Law / Political Science and Economics / Journalism / Management Law / Public Administration
 * College of Humanities and Sciences (1901-; Setagaya, Tokyo)
 * Philosophy / History / Japanese Language and Literature / Chinese Language and Culture / English Literature / German Literature
 * Sociology / Education / Physical Education / Psychology
 * Geography / Geosystem Sciences / Mathematics / Computer Science and System Analysis / Physics / Integrated Science in Physics and Biology / Chemistry
 * College of Economics (1904-; Chiyoda, Tokyo)
 * Economics (1st division / 2nd division / international course) / Industrial Management / Finance and Public Economics
 * College of Commerce (1904-; Setagaya, Tokyo)
 * Commerce / Business Administration / Accounting
 * College of Art (1921-; Nerima, Tokyo & Tokorozawa, Saitama)
 * Photography / Cinema / Fine Arts / Music / Literary Arts / Drama / Broadcasting / Design
 * College of International Relations (1978-; Mishima, Shizuoka)
 * International Relations / Intercultural Relations / Global Exchange Studies / International Business and Information
 * College of Science and Technology (1920-; Chiyoda, Tokyo & Funabashi, Chiba)
 * Civil Engineering / Transportation Engineering and Socio-Technology / Architecture / Oceanic Architecture and Engineering / Mechanical Engineering / Precision Machinery Engineering / Aerospace Engineering / Electrical Engineering / Electronics and Computer Science / Materials and Applied Chemistry / Physics / Mathematics
 * College of Industrial Technology (1952-; Narashino, Chiba)
 * Mechanical Engineering / Electrical and Electronic Engineering / Civil Engineering / Architecture and Architectural Engineering / Applied Molecular Chemistry / Industrial Engineering and Management / Mathematical Information Engineering / Liberal Arts and Basic Science
 * College of Engineering (1947-; Koriyama, Fukushima)
 * Civil Engineering / Architecture / Mechanical Engineering / Electrical and Electronics Engineering / Materials Chemistry and Engineering / Computer Science
 * School of Medicine (1925-; Itabashi, Tokyo)
 * Medicine
 * School of Dentistry (1921-; Chiyoda, Tokyo)
 * Dentistry
 * School of Dentistry at Matsudo (1971-; Matsudo, Chiba)
 * Dentistry
 * College of Bioresource Science (1943-; Fujisawa, Kanagawa)
 * Plant Science and Resources / Animal Sciences and Resources / Marine Sciences and Resources / Forest Sciences and Resources / Bioenvironmental and Agricultural Resources / Food Science and Technology / Agricultural and Biological Chemistry / Applied Biological Sciences / Food Economics / International Development Studies / Veterinary Medicine
 * College of Pharmacy (1952-; Narashino, Chiba)
 * Pharmacy / Biological Pharmacy
 * Correspondence Division (1948-; Chiyoda, Tokyo)

Graduate schools

 * Advanced Research Institute for the Sciences and Humanities
 * Graduate School of Law
 * Graduate School of Liberal Arts
 * Graduate School of Science and Technology
 * Graduate School of Integrated Basic Sciences
 * Graduate School of Economics
 * Graduate School of Commerce
 * Graduate School of Art
 * Graduate School of International Relations
 * Graduate School of Industrial Technology
 * Graduate School of Engineering
 * Graduate School of Medicine
 * Graduate School of Dentistry
 * Graduate School of Dentistry at Matsudo
 * Graduate School of Bioresource Science
 * Graduate School of Pharmacy
 * Graduate School of Business
 * Graduate School of Social and Cultural Studies
 * Law School

Teaching staff
See

Alumni
* Did not graduate.
 * Abe Shuji, President and CEO of Yoshinoya
 * Akio Mori, physiologist and writer
 * Akira Gomi, photographer
 * Banana Yoshimoto, writer
 * C. W. Nicol, environmentalist
 * Daishōmaru Shōgo, sumo wrestler
 * Daisuke Ono, voice actor
 * Takehiko Orimo, basketball player
 * Dong Biwu, Chinese communist revolutionary, Acting President of the People's Republic of China (1972–1975)
 * Endō Shōta, sumo wrestler
 * Hamanoshima Keishi, sumo wrestler
 * Hidenoumi Takuya, sumo wrestler
 * Higonoumi Naoya, sumo wrestler
 * Hiroko Mima, Miss Japan Universe and 15th runner-up for Miss Universe
 * Hiroshi Suga, photographer
 * Hiroshi Watanabe, photographer
 * Hiroshi Yamazaki, photographer
 * Hiroyuki Sanada, actor and martial artist
 * Ian Buruma, writer
 * Ichiro Ozawa, statesman*
 * Ishiura Masakatsu, sumo wrestler
 * Jōkōryū Takayuki, sumo wrestler
 * Jun Konno, judo wrestler
 * Junya Koizumi, statesman
 * Kōtarō Iizawa, photographic critic
 * Ken Domon, photographer*
 * Kishin Shinoyama, photographer
 * Kōichi Hamada, politician
 * Kotomitsuki Keiji, sumo wrestler
 * Kyuichi Tokuda, lawyer, politician, leader of the Japanese Communist Party (1945–1950)
 * Mainoumi Shūhei, sumo wrestler
 * Makoto Hasegawa, basketball player
 * Makoto Koga, statesman
 * Makoto Takimoto, judo wrestler
 * Masato Matsuura, CEO of Avex Group
 * Maximo Blanco, wrestler; professional Mixed Martial Artist in the UFC's Featherweight Division
 * Megumi Han, Voice actress and singer
 * Miyako Matsumoto, actress and professional wrestler
 * Remi Matsuo, musician and artist
 * Satoyama Kōsaku, sumo wrestler
 * Shoji Meguro, video game music composer
 * Sido Nakamura, actor*
 * Takamisakari Seiken, sumo wrestler
 * Takashi Sasano, actor*
 * Tetsuya Ichimura, photographer*
 * Tomoyuki Furumaya, film director
 * Toru Yano, professional wrestler
 * Toshiaki Kasuga, comedian
 * Tsutomu Sato, statesman
 * Wajima Hiroshi, sumo wrestler, the 54th yokozuna
 * Yasuaki Kurata, actor and martial artist
 * Yasuhiro Sonoda, statesman
 * Yoshikazu Tanaka, CEO and Founder of GREE Inc.
 * Yoshino Ōishi, photojournalist
 * Yoshinobu Nishizaki, animator
 * Yoshisada Yonezuka, 9th Dan (Kudan) Judo Instructor
 * Yoshiyuki Tomino, animator
 * Yui Hasegawa, footballer
 * Yutaka Takanashi, photographer
 * Yutaka Yaguchi, 9th Dan Shotokan Karate Master
 * Kei Tomiyama, voice actor*
 * Kentaro Miura, mangaka

Dirty tackle
In May 2018, at an American football game between the Nihon University Phoenix and the Kwansei Gakuin University Fighters, a Nihon University team member violently tackled an opposing team member from behind after play was over. The tackled Kwansei Gakuin team member received an injury in his back ligaments as a result, and had to take a month out from football to recover. A third-party committee convened by Nihon University to investigate the incident found that the Nihon University team's coach and assistant coach had instructed a team member to dirty-tackle that particular player on the Kwansei Gakuin team. Furthermore, another independent investigation, by the Kantoh Collegiate American Football Association, reached the same conclusion, and expelled the coaches from the Association. Nihon University reached an out-of-court settlement with the injured Kwansei Gakuin team player. A decision by Tokyo's Metropolitan Police Department not to prosecute the coaches was harshly criticized by those who had carried out the above investigations.

Drug possession
Three members of the Nihon University American football team were arrested between August and November 2023 for suspicion of using cannabis. In August 2023 police searched the team's dormitory after the university found pieces of a plant and pills there the previous month. The university suspended the team's activities in September following the first arrest. Following a second arrest in October, it was reported that notes from an August meeting of university executives suggested ten or more members of the American football team may have used cannabis.

In November 2023 the Nihon University Board of Trustees recommended the resignation of its President Takeo Sakai and its Vice President Yasuhiro Sawada, while Trustees Chair Mariko Hayashi agreed to a 50% pay cut. Soon after a third person was arrested, Japanese news sources reported that Nihon University planned to disband its American football team after a history of 83 years, during which it won the Koshien Bowl 21 times and the Rice Bowl four times. A third-party investigation into how the matter was handled, led by lawyer Hideaki Kubori, suggested a culture of secrecy and xenophobia on the part of university officials.

Alleged tax evasion by Chairman of the Board
On November 29, 2021, the former chairman of the board of Nihon University, Hidetoshi Tanaka, was arrested for alleged tax evasion, and indicted by prosecutors on December 20. Tanaka had allegedly received approximately 118 million yen in bribes from 2018 to 2020 paid by contractors in return for contracts being awarded for work at Nihon University. He had allegedly failed to declare the money as income, thus allegedly evading payment of approximately 52 million yen in tax. Tanaka announced his resignation on December 1, 2021. By accepting his resignation rather than dismissing him, the board ensured that Tanaka would remain eligible for a retirement bonus of hundreds of millions of yen.

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

title: Nihon University


text:
Mary Moorman

Mary Ann Moorman (born August 5, 1932) is an American woman who chanced to photograph US President John F. Kennedy a fraction of a second after he was fatally shot in the head in Dallas, Texas. The Badge Man, whom conspiracy theorists claim is one of Kennedy's assassins, is purportedly visible in another of her photographs taken that day.

Biography
Mary Ann Moorman was born Mary Ann Boshart. She married Donald G. Moorman in 1952 and divorced him in 1973. She later married Gary Krahmer in 1980.

Assassination witness
On November 22, 1963, U.S. President John F. Kennedy was assassinated in Dallas, Texas.

Moorman stated that her 11-year-old son had wanted to see Kennedy, but was unable to attend because of school. She said she promised to take a picture for him.

Moorman was standing on grass about 2 ft south of the south curb of Elm Street in Dealey Plaza, directly across from the grassy knoll and the North Pergola concrete structure that Abraham Zapruder and his assistant Marilyn Sitzman were standing on – during the assassination. Moorman stated that she stepped off the grass onto the street to take a photo with her Polaroid camera. Zapruder can be seen standing on the pergola in the Moorman photograph, with the presidential limousine already having passed through the line of sight between Zapruder and Moorman.

Both Moorman and her friend, Jean Hill, can be clearly seen in the Zapruder film. Between Zapruder frames 315 and 316, Moorman took a Polaroid photograph, her fifth that day, showing the presidential limousine with the grassy knoll area in the background.

Moorman's photograph captured the fatal head shot that killed President Kennedy. When she took it – approximately one sixth of a second after President Kennedy was struck in the head at Zapruder frame 313, Moorman was standing behind and to the left of President Kennedy, about 15 ft from the presidential limousine. Moorman said in a TV interview that immediately after the assassination, there were three or four shots close together, that shots were still being fired after the fatal head shot, and that she was in the line of fire. She later stated in a 2013 PBS documentary Kennedy Half Century that she was close enough to hear Jackie Kennedy exclaim that John had been shot.

In 2013, Moorman attempted to sell the original Polaroid through Cowan's Auctions in Cincinnati. The photo was expected to sell for between $50,000 and $75,000, but did not meet its reserve. It ultimately sold there. She had previously tried selling the photo to Sotheby's in New York, but the auction house deemed it to be "too sensitive to auction". That same year, she expressed her opinion on the assassination; she was convinced that Kennedy was killed as a result of a conspiracy. "I really don't know what exactly happened, but I do know there is bound to be a lot more to the story that hasn't been told," she said. "I was hoping it would come out in my lifetime, but who knows. So much has been hidden by the government; anything can take place and it can be hidden. Oswald probably wasn't a lone person, he probably had backers. I really do think it was a conspiracy".

Controversy
Whatever was captured in the background of Moorman's photo has been a matter of contentious debate. On the grassy knoll, some have claimed to identify as many as four different human figures, while others dismiss these indistinct images as either trees or shadows. Most often, one figure has been dubbed the "Badge Man" as it seems to resemble a uniformed police officer wearing a badge. Others claim to see Gordon Arnold, a man who claimed to have filmed the assassination from that area, a man in a construction hard hat, and a hatted man behind the stockade fence.

Moorman stated she heard a shot as the limousine passed her, then heard another two shots, "pow pow", when the president's head exploded. She stated that she could not determine where the shots came from, and that she saw no one in the area that appeared to have possibly been the assassin. Moorman was interviewed by the Dallas County Sheriff's Department and the FBI. She was called by the Warren Commission to testify, but due to a sprained ankle, she was unable to be questioned. She was never contacted by them again.

title: Mary Moorman


text:
Talk:Andreas Aagesen

Untitled
Why Andreas Aagesen is called Andrew Aagesen? Andres 10:57, 7 Dec 2003 (UTC)

title: Talk:Andreas Aagesen


text:
User:Mav/images/Lava Beds National Monument

Images that have homes
,


text:
Lepke Buchalter

Louis Buchalter, known as Louis Lepke or Lepke Buchalter, (February 6, 1897 – March 4, 1944) was a Jewish-American organized crime figure and head of the Mafia hit squad Murder, Inc., during the 1930s. Buchalter was one of the premier labor union racketeers in New York City during that era.

Charles Birger and Buchalter are the only National Crime Syndicate bosses to be executed after being convicted of murder. Buchalter was executed using the infamous "Old Sparky" electric chair after being sent "up the river" to Sing Sing Correctional Facility.

Background
Buchalter was born to a Russian-Jewish family in the Lower East Side neighborhood of the Manhattan borough of New York City on February 6, 1897. His mother, Rose Buchalter, called him "Lepkeleh" ("little Louis" in Yiddish), which later became "Lepke". Louis Buchalter had one sister and three brothers; one brother eventually became a dentist, another brother a college professor and rabbi, and the third brother a pharmacist. His father, Barnett Buchalter, was a Russian immigrant who operated a hardware store on the Lower East Side.

In 1909, when Buchalter was 12, his father died. In 1910, Buchalter finished elementary school and started a job selling theatrical goods. Reportedly, on the Lower East Side, he attended the Rabbi Jacob Joseph School, where he was an "honor roll" student. Soon after, his mother moved to Arizona for health reasons, leaving Buchalter in the care of his sister, Sarah. However, Buchalter was beyond her control.

On September 2, 1915, Buchalter was arrested in New York for the first time for burglary and assault, but the case was discharged.

In late 1915 or early 1916, Buchalter went to live with his uncle in Bridgeport, Connecticut. On February 29, 1916, Buchalter was arrested in Bridgeport on burglary charges and was sent to the Cheshire Reformatory for juvenile offenders in Cheshire, Connecticut, until July 12, 1917. After a dispute with his uncle over wages, Buchalter moved back to New York City.

On September 28, 1917, Buchalter was sentenced in New York to 18 months in state prison at Sing Sing in Ossining, New York, on a grand larceny conviction. After a transfer to Auburn Prison in Auburn, New York, Buchalter was released on January 27, 1919. On January 22, 1920, Buchalter returned to Sing Sing on a 30-month sentence for attempted burglary. He was released on March 16, 1922.

Rise to power
Upon Buchalter's 1922 release from prison, he started working with his childhood friend, mobster Jacob "Gurrah" Shapiro. Through force and fear, they began gaining control of the garment industry unions. Buchalter then used the unions to threaten strikes and demand weekly payments from factory owners while dipping into union bank accounts. Buchalter's control of the unions evolved into a protection racket, extending into areas such as bakery trucking. The unions were profitable for him and he kept a hold over them even after becoming an important figure in organized crime. Buchalter later formed an alliance with Tommy Lucchese, a leader of the Lucchese crime family, and together they controlled the garment district.

Buchalter and Shapiro moved into new and fashionable luxury buildings on Eastern Parkway (135) with family who were active synagogue goers (Union Temple and Kol Israel Synagogue of Brooklyn). In later years, Buchalter and his family lived in a penthouse in the exclusive Central Park West section of Manhattan.

In 1927, Buchalter and Shapiro were arrested for the murder of Jacob Orgen (Little Augie) and the attempted murder of Irish-American bootlegger Jack Diamond, a criminal rival. However, the charges were later dropped due to a lack of evidence.

Buchalter was described as a quiet man who for years managed to avoid the public spotlight. In conversations with his criminal associates, Buchalter preferred listening over talking. Buchalter generously compensated his gang members and took them to hockey games, boxing matches, and even winter cruises. On August 20, 1931, Buchalter married Betty Wasserman, a British-born widow of Russian descent, at New York City Hall. Buchalter adopted Betty's child from her previous marriage.

Murder, Inc.
In the early 1930s, Buchalter created an effective process for performing contract killings for Cosa Nostra mobsters; it had no name, but the press 10 years later called it Murder, Inc. The Cosa Nostra mobsters wanted to insulate themselves from any connection to these murders. Buchalter's partner, mobster Albert Anastasia, would relay a contract request from the Cosa Nostra to Buchalter. In turn, Buchalter would assign the job to Jewish and Italian street gang members from Brooklyn.

None of these contract killers had any connections with the major crime families. If they were caught, they could not implicate their Cosa Nostra employers in the crimes. Buchalter used the same killers for his own murder contracts. The Murder, Inc., killers were soon completing jobs all over the country for their mobster bosses.

In 1935, Buchalter arranged his most significant murder: the powerful New York gangster Dutch Schultz. Schultz had proposed to the National Crime Syndicate, a confederation of mobsters, that New York District Attorney Thomas Dewey be murdered. Co-founder of the Syndicate and leading mafioso Charles "Lucky" Luciano argued that a Dewey assassination would precipitate a massive law enforcement crackdown. An enraged Schultz said he would kill Dewey anyway and walked out of the meeting. After six hours of deliberations The Commission ordered Buchalter to eliminate Schultz. On October 23, 1935, Schultz was shot in a Newark, New Jersey tavern, and succumbed to his injuries the following day. In 1941, Buchalter killer Charles "The Bug" Workman was charged in the Schultz murder.

In 1935, law enforcement estimated that Buchalter and Shapiro had 250 men working for them, and that Buchalter was grossing over $1 million ($0 in current dollar terms) per year. They controlled rackets in the trucking, baking, and garment industries throughout New York. Buchalter also owned the Riobamba, a posh nightclub in Manhattan.

Downfall
On September 13, 1936, Murder, Inc. killers, acting on Buchalter's orders, gunned down Joseph Rosen, a Brooklyn candy store owner. Rosen was a former garment industry trucker whose union Buchalter took over. Rosen had angered Buchalter by refusing to leave town as Buchalter demanded when, despite the absence of proof, Buchalter believed Rosen was cooperating with District Attorney Thomas Dewey. At the time, no one was indicted in the Rosen murder.

On November 8, 1936, Buchalter and Shapiro were convicted of violating federal anti-trust laws in the rabbit fur industry in New York. While out on bail, both Buchalter and Shapiro disappeared. On November 13, both men were sentenced in absentia to two years in federal prison. The two men later appealed the verdict, but in June 1937 both convictions were upheld.

Manhunt
Before they could be taken into custody, both Buchalter and Shapiro disappeared. On November 9, 1937, the federal government offered a $5,000 reward for information leading to Buchalter's capture.

On December 1, 1937, the fugitive Buchalter was indicted in federal court on conspiracy to smuggle heroin into the United States. The scheme involved heroin hidden in the trunks of young women and couples traveling by ocean liner from China to France, then to New York City. Lepke bribed U.S. customs agents not to inspect the trunks.

On April 14, 1938, Shapiro surrendered to authorities in New York. However, Buchalter remained a fugitive.

Over the next two years, an extensive manhunt was conducted in both the United States and Europe, with reports of Buchalter hiding in Poland and Palestine. On July 29, 1939, Thomas Dewey requested that the City of New York offer a $25,000 reward for Buchalter's capture, citing a string of unsolved gangland murders. On August 24, 1939, Buchalter surrendered to FBI chief J. Edgar Hoover in front of a Manhattan hotel. The surrender deal was allegedly negotiated by the columnist and radio broadcaster Walter Winchell. It was later revealed that Buchalter had been hiding in New York City during his entire time as a fugitive.

After Buchalter was convicted on the federal narcotics trafficking charges, federal authorities turned him over to New York State for trial on labor extortion charges. On April 5, 1940, Buchalter was sentenced to 30 years to life in state prison on those charges. However, Buchalter was sent to Leavenworth Federal Penitentiary in Kansas to serve his federal sentence of 14 years for narcotics trafficking.

On August 20, 1940, Buchalter was indicted on murder charges in Los Angeles for the killing of Harry Greenberg, a mob associate of casino owner Meyer Lansky and mobster Bugsy Siegel. However, Buchalter never went to trial on this killing.

Murder trial
On May 9, 1941, Buchalter was arraigned in New York state court on the 1936 Rosen murder along with three other murders. Buchalter's order for the Rosen hit had been overheard by mobster Abe Reles, who turned state's evidence in 1940 and implicated Buchalter in four murders. Returned from Leavenworth to Brooklyn to stand trial for the Rosen slaying, Buchalter's position was worsened by the testimony of Albert "Tick-Tock" Tannenbaum. Four hours into deliberation, at 2 am on November 30, 1941, the jury found Buchalter guilty of first degree murder. On December 2, 1941, Lepke was sentenced to death along with his lieutenants Emanuel "Mendy" Weiss and Louis Capone. Buchalter's lawyers immediately filed an appeal.

In October 1942, the New York Court of Appeals voted four to three to uphold Buchalter's conviction and death sentence. (People v. Buchalter, 289 N.Y. 181) Two dissenting judges thought the evidence was so weak that errors in the jury instructions as to how to evaluate certain testimony were harmful enough to require a new trial. The third dissenter agreed, but added that, in his opinion, there was insufficient evidence to sustain a guilty verdict, so the indictment should be dismissed altogether (failure of proof means no retrial). Buchalter's lawyers then appealed to the U.S. Supreme Court.

The United States Supreme Court granted Buchalter's petition to review the case. In 1943, the Court affirmed the Buchalter conviction seven to zero, with two justices abstaining. (319 U.S. 427 (1943) ). His appeals were now exhausted.

Execution
When the U.S. Supreme Court confirmed Buchalter's conviction, he was serving his racketeering sentence at Leavenworth Federal Prison. New York State authorities demanded that the federal government turn over Buchalter for execution. On January 21, 1944, after many delays and much controversy, federal agents finally turned Buchalter over to state authorities, who immediately transported him to Sing Sing prison. Buchalter made several pleas for mercy, but they were rejected.

On March 4, 1944, Louis Buchalter was executed in the electric chair in Sing Sing. He had no final words. A few minutes before Buchalter's execution, his lieutenants Weiss and Capone were also executed.

Louis Buchalter was buried at the Mount Hebron Cemetery in Flushing, Queens.

In popular culture
During the late 1950s and early 1960s, Buchalter was portrayed by David J. Stewart in the 1960 film Murder, Inc.; by Gene Roth, Robert Carricart, and Joseph Ruskin in The Untouchables television series of 1959; as well as by John Vivyan and Shepherd Sanders in The Lawless Years television series.

The 1975 film Lepke, starring Tony Curtis, was based on Buchalter's life. Other portrayals include the 1981 film Gangster Wars by Ron Max.

Buchalter was also mentioned in "The Legend of Tennessee Moltisanti", episode eight of the first season of the popular HBO television series The Sopranos.

The poet Robert Lowell encountered Buchalter in prison during the 1940s, when Lowell was incarcerated for being a conscientious objector. Lowell described Buchalter (whom he calls "Czar Lepke") in his poem "Memories of West Street and Lepke", published in his book Life Studies (1959). Lowell's poem claims that prior to execution, Lepke had been lobotomized. Their meeting is discussed in the 1993 film Romeo Is Bleeding, where Lepke says he's in prison because he killed someone and Lowell says he's in prison because he didn't kill someone.

The Making of the Mob: New York (2015) – Buchalter was portrayed by Evan Boymel.

Buchalter's labor racketeering and extortion is prominently featured in Button Man,  a 2018 novel by Andrew Gross partially based on Gross's maternal grandfather's career in the New York City garment industry.

title: Lepke Buchalter


text:
Wilmersdorf

Wilmersdorf, an inner-city locality of Berlin, lies south-west of the central city. Formerly a borough by itself, Wilmersdorf became part of the new borough of Charlottenburg-Wilmersdorf in Berlin's 2001 administrative reform.

History
The village near Berlin was first mentioned in 1293 as Wilmerstorff, probably founded in the course of the German Ostsiedlung under the Ascanian margraves of Brandenburg. From the 1850s on Deutsch-Wilmersdorf was developed as a densely settled, affluent residential area, which in 1920 became a part of Greater Berlin. The former borough of Wilmersdorf included the localities of Halensee, Schmargendorf and Grunewald.

During the era of the Weimar Republic Wilmersdorf was a popular residential area for artists and intellectuals.

In 1923 the foundation stone for the first mosque in Germany was laid on the initiative of some Islamic students in Wilmersdorf. It was completed in 1925. The so called Wilmersdorfer Moschee (Mosque of Wilmersdorf) is still owned and maintained by the Lahore Ahmadiyya Movement.

In 1933, the year in which Hitler came to power, 13.5% of the population was Jewish; many of them were deported by the Nazis from Berlin-Grunewald Station. Deutsche Bahn established a memorial on 27 January 1998 at the historic track 17 ("Gleis 17"), where most of the deportation trains departed. The synagogue of Wilmersdorf in the Prinzregentenstraße was destroyed by the Nazis in the Reichspogromnacht on 9–10 November 1938. A memorial plaque commemorates the former synagogue. A new synagogue and community centre was established 2007 in the Münstersche Straße for the growing Jewish community in Wilmersdorf.

During World War II, Wilmersdorf was the location of a subcamp of the Sachsenhausen concentration camp. After 1945 Wilmersdorf was located in the British Zone of occupation.

Sights

 * Neo-Gothic Roman Catholic Saint Ludwig Church, 1897
 * Borough of the Rheingauviertel with the central Place Rüdesheimer Platz, 1910–1914
 * The historical subway stations on the line U3 from the times of the German Empire between Hohenzollerplatz and Rüdesheimer Platz, 1913
 * Friedhof Wilmersdorf
 * Ahmadiyya Mosque Berlin, Germany's oldest mosque from 1925
 * Artist Colony, built by the Guild of the German Stage, 1927
 * Schaubühne, famous theatre in the former Universum Cinema by Erich Mendelsohn, 1928
 * Kirche am Hohenzollernplatz by Ossip Klarwein and Fritz Höger, 1933
 * Russian Orthodox cathedral of the Resurrection of Jesus, 1938
 * Power station Berlin-Wilmersdorf, 1977

Higher education

 * Universität der Künste (Berlin University of Arts), Faculty of Music
 * IBZ Berlin, International Meeting Centre of Science

Primary and secondary schools

 * Comenius-Schule, a primary school, is in Wilmersdorf.
 * Halensee-Grundschule, a primary school, is in Halensee, near Wilmersdorf.
 * Svenska Skolan Berlin, Swedish School Berlin
 * Katholische Grundschule Sankt Ludwig, a catholic primary school
 * Nelson-Mandela-School, International School
 * Friedrich-Ebert-Gymnasium, a secondary school which is close to the Fennsee.
 * Goethe-Gymnasium, one of the most popular secondary schools in Berlin
 * Annie Heuser Schule, a private Waldorf education school
 * Annie Heuser Schule, a private Waldorf education school

Weekend schools

 * Zentrale Schule fur Japanisch Berlin e.V. (共益法人ベルリン中央学園補習授業校 Kyōeki Hōjin Berurin Chūō Gakuen Hoshū Jugyō Kō), is a weekend Japanese supplementary school. Established April 1997. The Japanische Ergänzungsschule in Berlin e.V. (ベルリン日本語補習授業校 Berurin Nihongo Hoshū Jugyō Kō), another weekend Japanese school, is held at Halensee-Grundschule.

Notable people

 * Maria von Maltzan German resistance against Adolf Hitler and the Nazi party, saved the lives of Jews in Berlin. Lived at Detmolder Straße 11, 1909–1997.
 * Paul Abraham, composer lived before he left Germany in 1933.
 * Jérôme Boateng, footballer for Bayern Munich and Germany, grew up in the area.
 * Heinz Berggruen, art dealer and founder of Museum Berggruen, grew up in the area
 * Berthold Brecht, poet, lived in Wilmersdorf with his partner Helene Weigel, until they left Germany in 1933.
 * Marlene Dietrich, actress, lived with her husband and her family in Wilmersdorf, before they finally left Germany in 1933.
 * Franz Pfemfert, published Die Aktion, the anti-nationalist, anti-militarist expressionist journal from premises at Nassauische Straße 17, 1911–1932.
 * Margarete Kahn, one of the first women to obtain a doctorate in Germany, Holocaust victim. Lived at 127 Rudolstädter Straße.
 * Erich Kästner, author and poet, lived in Wilmersdorf, while he wrote Emil and the Detectives, one of the most famous children's novels in Germany. The view out of his window with the colorful street scene at the Prager Platz was the inspiration for the book.
 * Eva Siewert, journalist, writer and activist, grew up in Wilmersdorf.
 * Hans Haustein (August 27, 1894 in Berlin – November 12, 1933 Ibid.) was a Jewish doctor and scientist in the Weimar Republic.
 * Konrad Zuse (June 22, 1910 – December 18, 1995), German inventor, born in Wilmersdorf

title: Wilmersdorf


text:
Groklaw

Groklaw is a website that covered legal news of interest to the free and open source software community. Started as a law blog on May 16, 2003, by paralegal Pamela Jones ("PJ"), it covered issues such as the SCO-Linux lawsuits, the EU antitrust case against Microsoft, and the standardization of Office Open XML.

Jones described Groklaw as "a place where lawyers and geeks could explain things to each other and work together, so they'd understand each other's work better".

Its name derives from "grok", roughly meaning "to understand completely", which had previously entered geek slang.

Other topics covered included software patents, DMCA, the actions of the RIAA against alleged illegal file sharers, and actions against free and open software such as Android and Linux.

Origins
According to a 2003 interview with Jones, the blog was started to cover legal news and to explain it to the tech community.

The first article was entitled "The Grokster Decision – Ode To Thomas Jefferson". It was about the effect of P2P on the music industry, and the then-recent court decision in MGM Studios, Inc. v. Grokster, Ltd., by Judge Steven Wilson in favor of the defendants. It also covered the previous Napster decision, and why it was different, causing Napster to be shut down. The article included a quote from Thomas Jefferson and references to David Boies, who was Napster's attorney.

The second post, on May 17, 2003, also covered legal issues – the SCO v. IBM lawsuit – entitled "SCO Falls Downstairs, Hitting its Head on Every Step". It criticized Caldera Systems for the way they were handling the suit outside of court, and included quotes from Bruce Perens, Richard Stallman, Steve Ballmer, and Linus Torvalds. It ended:
 * David Boies has agreed to represent SCO. I am trying to remind myself that our legal system is predicated on lawyers sometimes representing people they don't personally admire, and the system really does depend on someone being willing to take on unpopular clients. I know Boies doesn't use email, or at least he didn't the last time I checked. So maybe he doesn't quite get the tech ... ah, hang it all, there's no way around it: I feel bad he's chosen to represent them, especially after I posted an Ode singing his praises, and I hope he loses.

The blog soon became popular with the free software and open source communities and others, and attracted a community of volunteers and commenters. Its popularity caused it to outgrow Radio Userland, and on November 22, 2003, the standalone Groklaw website, hosted by ibiblio and running Geeklog software, was up and running.

Main focus
The main focus of Jones's writing became the Caldera Systems v. IBM litigation (Caldera Systems changed its name to The SCO Group during this time). Other issues were explored, including intellectual property and patent issues (for example, Microsoft IP claims against Linux, and the drafting of the GPL version 3). Groklaw was known for its contributors' ability to explain complex legal issues in simple terms and the research used in putting together articles. Members of the Groklaw community attended court hearings and interviewed movers and shakers in the software/IP world.

The site became a community effort. While Jones understood law, she was not a programmer. Many readers were techies, however, and when technical issues arose they provided relevant comments. This enabled Groklaw to solicit guest commentary on issues such as:

 * Linux Kernel coding practices
 * C Language programming
 * Operating systems programming
 * Operating systems history
 * Standards Organizations

Each of these issues appeared to have some application to the SCO v. IBM case, and most were revisited many times. Additional topics included later lawsuits by The SCO Group against Daimler Chrysler, Autozone, and Novell, the countersuit by Red Hat, and their implications and Microsoft's attempt to fast track OOXML as an International Organization for Standardization (ISO) standard.

Awards
Groklaw was cited by the attorneys for several firms in law journal articles. It also won awards:
 * 2012 – ABA Journal Blawg 100
 * 2010 – The Electronic Frontier Foundation (EFF) Pioneer Award
 * 2009 – Top 200 Tech Blogs: The Datamation 2009 List "The famed Groklaw is still going strong, far past the SCO case that first brought the blog to prominence."
 * 2008 – The Award for Projects of Social Benefit – The Free Software Foundation (FSF)
 * 2007 – Knowledge Masters Award for Innovation – Knowledge Trust and the Louis Round Wilson Academy
 * 2007 – Best FUD Fighter – Google-O'Reilly Open Source Awards
 * 2005 – Best News Site – ConsortiumInfo*.org – Pamela Jones/Groklaw: Best Community Site or Blog (Non-Profit)
 * 2005 – Best Blogger of the Year – Dana Blankenhorn, Corante
 * 2004 – Best Website of 2004 – The Inquirer
 * 2004 – Best Independent Tech Blog – TechWeb Network: Readers Choice Award
 * 2004 – Best Nontechnical or Community Website – Linux Journal: Editors' Choice Award
 * 2003 – Best News Site – OSDir.com: Editor's Choice Winner

Editorial stance
Groklaw was the personal creation of Jones, and it published articles (both news and opinion) from a self-described pro-FOSS, anti-FUD perspective.

While articles meticulously followed SCO's litigation activities, they were accompanied by reader-submitted comments that were "overwhelmingly pro-Linux and anti-SCO."

Media controversy
Jones was widely respected by journalists and people inside the Linux community. Steven J. Vaughan-Nichols wrote, "Jones has made her reputation as a top legal IT reporter from her work detailing the defects with SCO's case against IBM and Linux. Indeed, it is no exaggeration to say that her work has contributed enormously to everyone's coverage of SCO's cases."

Despite the high regard of Jones' peer journalists and the Linux community (or possibly in part because of it), a number of prominent attacks against Groklaw and Jones occurred. These attacks were documented and addressed in detail, on Groklaw and other web sites and also in court as part of the SCO litigation.

During the first week of May 2005, Maureen O'Gara, writing in Linux World, wrote an exposé claiming to unmask Jones. Two weeks before O'Gara's publication, McBride said that SCO was investigating Jones' identity. The article included alleged, but unverified, personal information about Jones, including a photo of Jones' supposed house and purported addresses and telephone numbers for Jones and her mother. After a flood of complaints to the publisher, lobbying of the site's advertisers, and claims of a denial-of-service attack launched against the Sys-Con domain, Linux Business News ' publisher Sys-Con issued a public apology, and said they dropped O'Gara and her LinuxGram column. Despite this assertion, O'Gara remained with Sys-Con; as of 2009, she is the Virtualization News Desk editor at Sys-Con Media, who describe her as "[o]ne of the most respected technology reporters in the business" and has her work published in multiple magazines owned by Sys-Con Media.

SCO executives Darl McBride and Blake Stowell also denigrated Jones, and claimed that she worked for IBM. Jones denied this allegation, as did IBM in a court filing. During an SCO conference call on April 13, 2005, McBride said, "The reality is the web site is full of misinformation, including the people who are actually running it" when talking about Groklaw, adding also "What I would say is that it is not what it is purported to be". Later developments in the court cases showed that McBride's statements to the press regarding the SCO litigation had limited credibility; very few such statements were ever substantiated and most were shown to be false. For example, McBride claimed that SCO owned the copyrights to UNIX, and SCO filed suit to try to enforce these claims. The outcome went against McBride's claims. The jury found that SCO had not purchased these copyrights. SCO appealed this ruling and lost. McBride also made a claim to the press that there was a "mountain of code" misappropriated to create Linux. When SCO finally presented their evidence of infringement, which centered on nine lines of error name and number similarities in the file errno.h, Judge Wells famously said "Is this all you've got?" Professor Randall Davis of MIT later made a convincing demonstration that there were no elements of UNIX which might be copyright protectable present in the Linux source code.

Additional projects
Anticipating further legal threats against GNU, Linux, and the free software community, Jones launched Grokline, a Unix ownership timeline project, in May 2004. One notable result of the Groklaw/Grokline effort was obtaining and publishing the 1994 settlement in USL v. BSDi, which for over a decade had been sealed by the parties. The document was obtained through a California freedom of information statute (the University of California, being a publicly funded institution, is required by law to make almost all of its documents public), and the release of the settlement answered many questions as to the ownership of the Unix intellectual property.

The Linux documentation project Grokdoc wiki was started in 2004, with the stated goal "to create a useful manual on basic tasks that new users will find simple and clear and easy to follow."

Groklaw extensively covered patent problems with software and hardware, use of the DMCA against free software ideals, Open standards, DRM, GPLv3, and published The Daemon, the GNU & the Penguin, a series of articles by Peter Salus covering the history of Unix, Linux and the GNU project.

It covered the Oracle v. Google in which Oracle alleged that Google's Android platform infringed copyrights and patents related to Java.

Later history
In January 2009, Groklaw entered a second phase, focusing on consolidation and cleanup of the legal history collected on the site.

In April 2010, Groklaw was selected by the Library of Congress for its web archival project, in the category of Legal Blogs.

On April 9, 2011, Jones announced that Groklaw would stop publishing new articles on May 16, 2011, its 8th anniversary, as it had accomplished its original mission of revealing the truth behind the SCO lawsuits.

On May 16, 2011, Jones reaffirmed her desire to step down from writing daily articles and announced that the new editor would be Mark Webbink.

Subsequent to this decision, new patent and copyright based attacks on the Android operating system led to Jones resuming an editorial role, and along with Mark Webbink she moderated and edited the site.

On August 20, 2013, a final article appeared on Groklaw, explaining that due to pervasive government monitoring of the Internet, there could no longer be an expectation of the sort of privacy online that was necessary to collaborate on sensitive topics. Citing the closure of Lavabit earlier that month, Jones wrote "I can't do Groklaw without your input.... and there is now no private way, evidently, to collaborate." and "What I do know is it's not possible to be fully human if you are being surveilled 24/7... I hope that makes it clear why I can't continue. There is now no shield from forced exposure."

During 2020, the site was intermittently unavailable. , the home page and parts of the content are still available.

title: Groklaw


text:
Talk:Jaguar Cars

Brand ownership
As with the Land Rover article, I've tried to align the brand ownership content (prose and infobox) with what I have been able to glean from the UK trademark registry. The records are not complete, so some interpolation might have crept in, but that website is surely the most reliable source available for this information. Any ideas about how this information might be improved upon and made more accurate or reflective of the reality would be appreciated. -- DeFacto (talk). 23:12, 26 January 2018 (UTC)

Ranger rover is under tata group so jaguar also under the tata group Ritik saini4477 (talk) 12:18, 18 September 2022 (UTC)

Current Cars List
Since the XJ is no longer being produced or sold it should be removed from the list and the XF-L should be added to the list in one way or another
 * I'm not that aware of the Jaguar range, so another editor may give you more feedback on that. But, if you think the list is wrong you may be bold and fix it. Be aware of WP:BRD. --Urbanoc (talk) 17:30, 12 April 2021 (UTC)

Jaguar Cars did not go defunct in 2012
Hello, The article provides wrong information about Jaguar Cars going defunct in 2012. In reality, it was just renamed to Jaguar Land Rover Limited. Here is the company record in the official register - see the previous names section at the bottom, and here is a document confirming the name change. Jaguar Land Rover should not have a separate article, because it is the same company. Alternatively, this article may be cleaned up to represent the brand only, and the company info should be moved to "history" in Jaguar Land Rover. — Preceding unsigned comment added by 91.217.58.11 (talk) 18:52, 10 November 2021 (UTC)

 * Jaguar Cars is no longer a car manufacturing company, all Jaguar and all Land Rover cars have been manufactured by Jaguar Land Rover since 1 January 2013. This article covers the use of the Jaguar marque from its inception to the present day, as well as the Jaguar Cars company until it stopped making cars at the end of 2012. -- DeFacto (talk). 14:49, 18 September 2022 (UTC)

Semi-protected edit request on 7 July 2023
The wrong tense has been used in relation to the sales of the F-Type stating that "Sales will begin in 2013" Alex Argent (talk) 10:38, 7 July 2023 (UTC)

 * Apologies. I meant the commencing of sales in 2013, not production. Alex Argent (talk) 10:48, 7 July 2023 (UTC)
 * it will help if you can provide a reliable sources that states when sales began. M.Bitton (talk) 15:53, 7 July 2023 (UTC)
 * Red information icon with gradient background.svg Not done: please provide reliable sources that support the change you want to be made. Throast  { { ping }} me! (talk | contribs) 14:04, 8 July 2023 (UTC)

Reason for protection
What’s the deal with this page being protected? It’s just a page about a car what’s the problem with it? 2601:152:4A7F:8080:8108:5748:3805:52FE (talk) 23:57, 30 April 2024 (UTC)

title: Talk:Jaguar Cars