Maitreyee Dutta

Elena Panteley

Antonio Loría

Srikant Sukumar

MRAC-based dynamic consensus of linear systems with biased measurements, over directed networks

Keywords: Measurement bias, directed graphs, multi-agent systems, linear systems, Lyapunov stability

Dynamic consensus pertains to the achievement of a common collective behavior among interconnected systems. In this paper we address the dynamic consensus control problem for generic linear systems (stable, unstable, marginally stable, . . . ) interconnected over directed networks and under the influence of biased measurements. Essentially, the control problem consists in redesigning a standard distributed consensus controller which, for each system, relies on own state biased measurements and respective erroneous data received from a set of neighbors. The difficulty in such a scheme resides in the fact that the measurement bias is directly amplified by the control gain so it cannot be handled as an additive external disturbance. Instead, our control design relies, on one hand, on the solution of a Riccati equation and, on the other, on the design of an estimator reminiscent of a model-reference-adaptive control design. The estimator successfully computes a bias estimate and completely compensates for its effect if the bias is constant-indeed, in this case, we establish exponential stability of the consensus manifold and we show that the controller provides robustness with respect to time-varying biases.

Introduction

It is not a platitude to say that feedback is the essence of automatic control, so using faulty sensor data can have major repercussions in the closed-loop stability of the physical systems and can lead to severe accidentssee [START_REF] Lombaerts | Fault tolerant flight control, a physical model approach[END_REF]. Sensor faults may consist, e.g., in bias, drift, scaling, noise, etc. [START_REF] Balaban | Modeling, detection, and disambiguation of sensor faults for aerospace applications[END_REF]. Measurement bias, which is the one of interest here, is a persistent offset in the reading, that may affect all the samples gathered in a particular deployment.

In the context of networked multi-agent systems, while a bidirectional communication helps in decoupling the sensor biases from the state measurements, as seen in e.g., [START_REF] Shi | Bias estimation in sensor networks[END_REF], [START_REF] Sinha | Consensus of networked double integrator systems under sensor bias[END_REF], the influence of one agent over its neighbor is not always reciprocal in nature, so decoupling of the bias from the measurements can turn out to be an intricate problem in directed-topology networks-see e.g, [START_REF] Sukumar | On consensus of double integrators over directed graphs and with relative measurement bias[END_REF], [START_REF] Meng | Adaptive consensus for heterogeneous multi-agent systems under sensor and actuator attacks[END_REF], [START_REF] Cao | Event-triggered control for multiagent systems with sensor faults and input saturation[END_REF], [START_REF] Tan | Neuroadaptive asymptotic consensus tracking control for a class of uncertain nonlinear multiagent systems with sensor faults[END_REF], [START_REF] Borzone | Hybrid formalism for consensus of a general class of multi-agent systems with biased measurements[END_REF], [START_REF] Yan | Formation consensus for discretetime heterogeneous multi-agent systems with link failures and actuator/sensor faults[END_REF]. Different techniques are applied in the latter references to cope with constant [START_REF] Sukumar | On consensus of double integrators over directed graphs and with relative measurement bias[END_REF] or time-varying bias [START_REF] Meng | Adaptive consensus for heterogeneous multi-agent systems under sensor and actuator attacks[END_REF]: adaptive control in [START_REF] Meng | Adaptive consensus for heterogeneous multi-agent systems under sensor and actuator attacks[END_REF], event-triggered control in [START_REF] Cao | Event-triggered control for multiagent systems with sensor faults and input saturation[END_REF], adaptive neural control in [START_REF] Tan | Neuroadaptive asymptotic consensus tracking control for a class of uncertain nonlinear multiagent systems with sensor faults[END_REF], and an LMI-based approach in [START_REF] Yan | Formation consensus for discretetime heterogeneous multi-agent systems with link failures and actuator/sensor faults[END_REF]. In [START_REF] Borzone | Hybrid formalism for consensus of a general class of multi-agent systems with biased measurements[END_REF] a distributed piece-wise constant impulsive reference is used.

A general connected directed graph may consist in, or contain, a spanning tree, so leader-follower consensus is applicable if the communication topology is a spanning tree with the root node behaving as a leader. In [START_REF] Meng | Adaptive consensus for heterogeneous multi-agent systems under sensor and actuator attacks[END_REF], [START_REF] Cao | Event-triggered control for multiagent systems with sensor faults and input saturation[END_REF], and [START_REF] Tan | Neuroadaptive asymptotic consensus tracking control for a class of uncertain nonlinear multiagent systems with sensor faults[END_REF] the leader-follower consensus problem is addressed, under the assumption that the measurement of the follower state/output variables are unreliable but the measurement of the leader states is bias-free. However, such assumption appears conservative, specially in a leaderless consensus scenario.

A significant difficulty to deal with measurement bias is that, in general, it cannot be dealt with using ad hoc robust control techniques tailored to cope with additive disturbances and consisting, essentially, in increasing the gain-the bias, if anything, is amplified by the control gain. In [START_REF] Sukumar | On consensus of double integrators over directed graphs and with relative measurement bias[END_REF] the leaderless consensus problem is tackled using measurements affected by constant bias; the synchronization error is shown to be ultimately ultimately bounded (UUB).

In general, bias cannot be dealt with as a common uncertainty in lumped parameters either. Yet, some works borrow inspiration from the classical model-reference adaptive control (MRAC) method-see e.g., [START_REF] Ioannou | Robust adaptive control[END_REF] to successfully design distributed model-reference adaptive controllers for cooperative tracking [START_REF] Peng | Distributed model reference adaptive control for cooperative tracking of uncertain dynamical multi-agent systems[END_REF], [START_REF] Sumizaki | Adaptive consensus on a class of nonlinear multi-agent dynamical systems[END_REF], [START_REF] Liu | Adaptive leader-following consensus control of multi-agent systems using model reference adaptive control approach[END_REF], as well as for cooperative regulation [START_REF] Mei | A unified framework for adaptive leaderless consensus of uncertain multiagent systems under directed graphs[END_REF], and for different communication topologies-whether it is undirected [START_REF] Peng | Distributed model reference adaptive control for cooperative tracking of uncertain dynamical multi-agent systems[END_REF], [START_REF] Sumizaki | Adaptive consensus on a class of nonlinear multi-agent dynamical systems[END_REF], [START_REF] Liu | Adaptive leader-following consensus control of multi-agent systems using model reference adaptive control approach[END_REF] or directed [START_REF] Mei | A unified framework for adaptive leaderless consensus of uncertain multiagent systems under directed graphs[END_REF].

The application of MRAC-inspired techniques for leader-follower consensus generally relies on assimilating the leader node as the singular reference modele.g., [START_REF] Peng | Distributed model reference adaptive control for cooperative tracking of uncertain dynamical multi-agent systems[END_REF], [START_REF] Sumizaki | Adaptive consensus on a class of nonlinear multi-agent dynamical systems[END_REF], [START_REF] Liu | Adaptive leader-following consensus control of multi-agent systems using model reference adaptive control approach[END_REF]. In [START_REF] Mei | A unified framework for adaptive leaderless consensus of uncertain multiagent systems under directed graphs[END_REF], a framework for adaptive leaderless consensus of linear (homogeneous) agents with uncertain dynamics is proposed. The method consists in designing a reference model for each system and having the latter track the output generated by the corresponding linear reference model with relative state measurement as input.

Now, in all of the above cited references, MRACbased techniques are used to deal with uncertain systems, but none of them considers biased relative state measurements. In the continuous-time domain, the study of leader-less dynamical consensus among agents exchanging biased state measurements (even for root nodes) through unidirectional network is rather limited. In [START_REF] Patre | Accommodating sensor bias in MRAC for state tracking[END_REF] an MRAC-based technique is proposed to accommodate constant sensor bias, but not in a multi-agent context.

Thus, the dynamic (leaderless) consensus problem problem under measurement bias for multi-agent systems interconnected over generic directed graphs is rather limited. In this paper, we address it for generic identical linear systems under the assumption that measurement bias is different and bounded for each agent. More precisely, our contributions are the following:

(1) in contrast to [START_REF] Peng | Distributed model reference adaptive control for cooperative tracking of uncertain dynamical multi-agent systems[END_REF], [START_REF] Sumizaki | Adaptive consensus on a class of nonlinear multi-agent dynamical systems[END_REF], [START_REF] Liu | Adaptive leader-following consensus control of multi-agent systems using model reference adaptive control approach[END_REF], and [START_REF] Mei | A unified framework for adaptive leaderless consensus of uncertain multiagent systems under directed graphs[END_REF] we provide an MRAC-based control law for leader-less consensus among agents, which can override the effect of biased measurements; (2) in contrast to [START_REF] Meng | Adaptive consensus for heterogeneous multi-agent systems under sensor and actuator attacks[END_REF], [START_REF] Cao | Event-triggered control for multiagent systems with sensor faults and input saturation[END_REF], and [START_REF] Tan | Neuroadaptive asymptotic consensus tracking control for a class of uncertain nonlinear multiagent systems with sensor faults[END_REF], we do not assume that the leader (root) nodes are equipped with biasfree sensors; (3) unlike [START_REF] Sukumar | On consensus of double integrators over directed graphs and with relative measurement bias[END_REF], in which the synchronization error is only uniformly ultimately bounded in presence of constant biases, we establish sufficient conditions for exponential stability of the synchronization manifold in the presence of constant biases and UUB if the latter are time-varying. The remainder of the paper is organized as follows. In the next section we provide a detailed problem formulation, in Section 3 we present our main results; in Section 4 we provide some simulation results to illustrate our theoretical findings and we conclude with some closing remarks in Section 5.

Problem Formulation: System Setup

We consider N ∈ N identical, linear, autonomous systems,

ẋi = Ax i + Bu i , ∀i = {1, 2, • • • , N }, (1)
where x i ∈ R n , u i ∈ R p is the control input and A, B are constant matrices of compatible dimensions satisfying the following: (A1) the pair (A,B) is stabilizable, (A2) the system matrix A is of full rank.

For these systems, we address the dynamic consensus control problem. This pertains to making all systems achieve a common dynamic behavior, hence, such that lim

t→∞ |x i (t) -x j (t)| → 0, ∀ i ̸ = j ∈ {1, 2, • • • , N }, (2)
via a distributed consensus controller of the form

u i = K j∈Ni a ij (x i -x j ) + ν i . (3)
In the expression above, the first term on the righthand side corresponds to a common consensus control law, preceded by a suitable gain matrix K ∈ R p×n to be determined. It is assumed that the states x i are not measurable, or, more precisely, that (A3) the i th agent has access to its own state and that of its neighbors, but these measurements are biased by an unknown offset δ i , that is, xi := x i + δ i ; (A4) the bias δ i is either constant or time-varying with bounded derivative. The term ν i in (3) denotes an additional, redesign, control input to be determined, notably to compensate for the effect of the measurement bias.

Finally, the coefficients a ij ≥ 0 represent the existence (if a ij > 0) or the absence (if a ij = 0) of a one-way interaction from the j th node to the i th node. As it is customary, we consider that there are no self-loops, so a ii = 0. More precisely, we assume that (A5) the network of interconnections form a directed connected graph with Laplacian matrix

L = [l ij ] ∈ R N ×N
, where

l ii = j∈Ni a ij , l ij = -a ij . (4)
Using the interactions as per (4), the i th node can also have access to the bias estimates δj of the state measurement's unknown bias δ j ∀j ∈ N i . The Laplacian, which is positive semidefinite in the case that the graph is undirected and connected, may be used with relative easiness to construct Lyapunov functions that serve to analyze the stability of the consensus manifold {x i = x j }.

For connected directed networks L is non-symmetric and this hinders the construction of suitable Lyapunov function candidates. To that end, we recall the following useful Lemmata.

Lemma 1 [START_REF] Ren | Consensus seeking in multiagent systems under dynamically changing interaction topologies[END_REF][START_REF] Ren | Distributed consensus in multivehicle cooperative control[END_REF] If a directed network has a directed spanning tree, then the Laplacian matrix L = [l ij ] ∈ R N ×N has a singular zero eigenvalue and N -1 eigenvalues have strictly positive real part. That is,

σ 1 (L) = 0, ℜ{σ k (L)} > 0, k = {2, • • • , N }.
On the other hand, the right eigenvector of the zero eigenvalue is given by [START_REF] Panteley | Strict lyapunov functions for consensus under directed connected graphs[END_REF] Let us consider a directed graph G of order N containing a spanning tree and its Laplacian matrix is L ∈ R N ×N . Then, for any positive definite symmetric matrix Q L ∈ R N ×N and α ∈ R + , there exists another positive definite symmetric matrix P L ∈ R N ×N such that

1 N = [1 1 • • • 1] ⊤ and the left eigenvec- tor v l satisfies N k=1 v l k = 1 and v ⊤ l L = 0 ⊤ N . □ Lemma 2
P L L + L ⊤ P L = Q L -α[P L 1 N v ⊤ l + v l 1 ⊤ N P L ]. (5)
□ 3 Main results

Control design

The consensus controller proposed in this paper relies on a distributed estimator whose design, along similar lines as those in [START_REF] Peng | Distributed model reference adaptive control for cooperative tracking of uncertain dynamical multi-agent systems[END_REF], [START_REF] Sumizaki | Adaptive consensus on a class of nonlinear multi-agent dynamical systems[END_REF], [START_REF] Liu | Adaptive leader-following consensus control of multi-agent systems using model reference adaptive control approach[END_REF], and [START_REF] Mei | A unified framework for adaptive leaderless consensus of uncertain multiagent systems under directed graphs[END_REF], borrows inspiration from model-reference adaptive control. The control strategy consists in coupling each agent's dynamics to that of an estimator of the form

ẋi =Ax i + ϵBF j∈Ni a ij (x i -x j ) -ϵBF j∈Ni a ij ( δi -δj ) (6)
where the coupling weight ϵ > 0, the control gain F ∈ R p×n , and the bias estimate δi are yet to be defined. Then, to achieve the consensus control objective (2) the first control goal is set to make the estimators (6) achieve dynamic consensus, that is,

lim t→∞ |x i (t) -xj (t)| → 0, ∀ i ̸ = j ∈ {1, 2, • • • , N } (7)
and, second, to steer each agent's state trajectories to its corresponding estimator's, that is,

lim t→∞ |x i (t) -xi (t)| → 0 ∀ i ∈ {1, 2, • • • , N }. ( 8
)
Note that ( 7) and ( 8) imply (2) since

{x i = xj } ∩ {x i = xi } ∀ i, j ∈ {1, 2, • • • , N } implies that {x i = x j } for all i ∈ {1, 2, • • • , N }.
However, since the states x i are not measurable, we pose as objective to steer the measured state xi to the corresponding biased state of the reference model in [START_REF] Ioannou | Robust adaptive control[END_REF], that is, to make ẽi → 0, where

ẽi = x i -(x i + δi ). ( 9
)
It is remarked that this task, on one hand, appears feasible since ẽi is available and, on the other hand, is useful

since ẽi = x i -xi + δi , (10)
with δi := δ i -δi , so (8) holds if ẽi → 0 and δi is updated so as to have

lim t→∞ | δi (t) -δ i | → 0, ∀ i ∈ {1, 2, • • • , N }. (11)
To steer ẽi → 0 the control law u i in ( 12) is redesigned as follows. We start by differentiating on both sides of [START_REF] Liu | Adaptive leader-following consensus control of multi-agent systems using model reference adaptive control approach[END_REF]

to obtain ėi = Aẽ i -A δi + Bu i + δi -ϵBF j∈Ni a ij (x i -x j ) + ( δi -δj ) , (12)
for which we used (1) and

ẋi = Ax i + ϵBF j∈Ni a ij (x i -x j ) + ( δi -δj ) . ( 13
)
First, note that the terms on the second line of ( 12) can be canceled directly since they correspond exactly to the data available to the i th node. Second, we may insert a control term of the form F ẽi , such that (A + BF ) is Hurwitz; this is possible since the pair (A, B) is stabilizable by assumption. More precisely, the latter implies the existence of a matrix M = M ⊤ > 0 that solves the algebraic Riccati equation

M A + A ⊤ M -M BB ⊤ M = -Q, (14)
for any given Q = Q ⊤ > 0. Thus, for any such given Q that generates M as above, we define F = -B ⊤ M -cf. [START_REF] Xiao | Adaptive consensus in leader-following networks of heterogeneous linear systems[END_REF][START_REF] Yang | Decentralized eventtriggered consensus for linear multi-agent systems under general directed graphs[END_REF][START_REF] Chen | Event-triggered coordination of multi-agent systems via a lyapunov-based approach for leaderless consensus[END_REF][START_REF] Li | Consensus of multi-agent systems with general linear and lipschitz nonlinear dynamics using distributed adaptive protocols[END_REF][START_REF] Li | Designing fully distributed consensus protocols for linear multi-agent systems with directed graphs[END_REF] and we redefine u i in (3) as

u i = ϵF j∈Ni a ij (x i -x j ) -ϵF j∈Ni a ij ( δi -δj ) + F ẽi .
(15) Then, using [START_REF] Panteley | Synchronization and dynamic consensus of heterogeneous networked systems[END_REF] in [START_REF] Loría | Uniform exponential stability of linear time-varying systems: revisited[END_REF], we obtain

ėi =(A + BF )ẽ i -A δi + δi . (16)
Since (A+BF ) is Hurwitz by design, it is left to define the dynamics of the bias estimates δi -hence that of δiso as to ensure that δi and δi vanish asymptotically. The design of the bias estimation law depends on whether δ i is constant or time-varying, as per Assumption (A4), and is explained in the next sections.

Dynamic consensus under constant bias

In the case that δ i is constant we define δi = -M -1 A ⊤ M ẽi , (17)
where M solves ( 14) for any given positive definite Q. Such estimation law is distributed since it depends only on ẽi , which is defined by variables that pertain to the i th agent only-see [START_REF] Li | Designing fully distributed consensus protocols for linear multi-agent systems with directed graphs[END_REF]. Then, we have the following. Remark 3 (Notation) In the remainder of the paper, we use the notations σ H m and σ H M to represent the minimum and maximum eigenvalue of any square matrix H respectively.

• Proposition 4 (Dynamic consensus) Consider N identical linear systems as in (1) in closed loop with the distributed control input given by [START_REF] Panteley | Synchronization and dynamic consensus of heterogeneous networked systems[END_REF], with ϵ ≥ σ P L M where P L solves (5) for Q L = I N , the feedback matrix F = -B ⊤ M where M solves (14), the reference model [START_REF] Ioannou | Robust adaptive control[END_REF], and the bias update law [START_REF] Patre | Accommodating sensor bias in MRAC for state tracking[END_REF]. Then, under Assumptions (A1)-(A5), (2) holds.

□ The previous statement follows after a cascades argument and the rationale presented previously. The limit in (2) holds if so do those in [START_REF] Khalil | Nonlinear systems[END_REF] and [START_REF] Li | Consensus of multi-agent systems with general linear and lipschitz nonlinear dynamics using distributed adaptive protocols[END_REF]. The latter is established in Proposition 5, which is presented next, and ( 7) is established in Proposition 7 farther below. Proposition 5 (bias estimation) Consider N identical linear systems as in (1) in closed loop with (15), [START_REF] Ioannou | Robust adaptive control[END_REF], and (17) under the conditions of Proposition 4. Then, there exist λ and κ > 0, such that

|ξ(t)| ≤ κ|ξ(t • )|e -λ(t-t•) , ∀t ≥ t • , (18)
where ξ := ẽ⊤ δ⊤ ⊤ , ẽ⊤ :

= [ẽ ⊤ 1 • • • ẽ⊤ N ], and δ⊤ := [ δ⊤ 1 • • • δ⊤ N ].
In turn, the limits in (8) and (11) hold. □ Proof. In compact form, the multi-agent closed-loop equations correspond to ė =

I N ⊗ (A + BF ) ẽ -[I N ⊗ A] δ, + I N ⊗ (M -1 A ⊤ M ) ẽ (19a) δ = I N ⊗ (M -1 A ⊤ M ) ẽ. (19b)
Then, consider the Lyapunov function candidate

V (ẽ, δ) = ẽ⊤ (I N ⊗ M )ẽ + δ⊤ (I N ⊗ M ) δ, (20)
which, since M is symmetric positive definite, is both positive definite and decrescent. Indeed, V satisfies

σ M m |ξ| 2 ≤ V (ẽ, δ) ≤ σ M M |ξ| 2 . ( 21
)
Furthermore, in virtue of ( 14), the time-derivative of V along the trajectories of ( 19) is negative semidefinite. Indeed,

V (ẽ, δ) = ẽ⊤ I N ⊗ (M A + A ⊤ M -2M BB ⊤ M ) ẽ -2ẽ ⊤ [ I N ⊗ (M A) ] δ + 2ẽ ⊤ [ I N ⊗ (A ⊤ M ) ]ẽ + 2 δ⊤ [ I N ⊗ (A ⊤ M ) ]ẽ = 2ẽ ⊤ I N ⊗ (M A + A ⊤ M -M BB ⊤ M ) ẽ = -2ẽ ⊤ [I N ⊗ Q]ẽ ≤ 0. ( 22
)
From ( 21)-( 22), it follows that the origin, {(ẽ, δ) = (0, 0)}, for the system (19a)-( 19b) is uniformly globally stable. In particular, it is uniformly stable and all solutions are uniformly globally bounded.

To establish uniform global exponential stability we use the following statement, which follows from [12, Lemma 3]. Lemma 6 Consider the system ξ = f (t, ξ), where f (•, ξ) is locally integrable and f (t, •) is locally Lipschitz uniformly in t. Assume that there exist constants p, and c > 0 such that the solution ξ(

•, t • , ξ • ) of ξ = f (t, ξ) sat- isfies max sup t≥t• |ξ(t)|, ∞ t• |ξ(τ )| p 1/p dτ ≤ c|ξ • |, (23)
for all initial conditions3 (t • , x • ) ∈ R ≥0 × R n . Then, the origin {ξ = 0} is uniformly globally exponentially stable and, moreover, (18) holds with κ = ce 1/p and λ = [pc p ] -1 . □ Thus, the rest of the proof consists in establishing the bounds in [START_REF] Sukumar | On consensus of double integrators over directed graphs and with relative measurement bias[END_REF] for the closed-loop system [START_REF] Ren | Consensus seeking in multiagent systems under dynamically changing interaction topologies[END_REF]. The first inequality in [START_REF] Sukumar | On consensus of double integrators over directed graphs and with relative measurement bias[END_REF] follows by integrating along trajectories on both sides of the inequality V ≤ 0 in [START_REF] Sinha | Consensus of networked double integrator systems under sensor bias[END_REF] and using [START_REF] Shi | Bias estimation in sensor networks[END_REF] to obtain

V (ẽ(t), δ(t)) ≤ V (ẽ(t • ), δ(t • )) (24) σ M m |ξ(t)| 2 ≤ V (ẽ(t), δ(t)) ≤ σ M M |ξ(t • )| 2 , ∀ t ≥ t • ≥ 0, which, in turn, implies that |ξ(t)| ≤ σ M M σ M m |ξ(t • )|, ∀ t ≥ t • , (25)
so the first inequality in [START_REF] Sukumar | On consensus of double integrators over directed graphs and with relative measurement bias[END_REF] holds with p = 2 and c 2 := σ M M /σ M m .

On the other hand, integrating on both sides of V = -2ẽ ⊤ (I N ⊗Q)ẽ in [START_REF] Sinha | Consensus of networked double integrator systems under sensor bias[END_REF], along the trajectories, we obtain

V (ẽ(t), δ(t)) -V (ẽ(t • ), δ(t • )) ≤ -2σ Q m t t• |ẽ(τ )| 2 dτ.
Rearranging the letter and using [START_REF] Sumizaki | Adaptive consensus on a class of nonlinear multi-agent dynamical systems[END_REF] gives

2σ Q m t t• |ẽ(τ )| 2 dτ ≤ 2V (ẽ(t • ), δ(t • ))
which, in view of [START_REF] Shi | Bias estimation in sensor networks[END_REF], implies that

lim t→∞ t t• |ẽ(τ )| 2 dτ ≤ σ M M σ Q m |ξ(t • )| 2 . ( 26
)
The latter establishes the second inequality in ( 23), for the ẽ-part of ξ. To compute a similar inequality for δ, consider the function W : R 2nN → R, defined by

W (ξ) = ẽ⊤ [I N ⊗ A] δ (27)
and let σ A M > 0 be such that |A| ≤ σ A M . Then, along the trajectories,

W (ξ(t)) ≤ σ A M 2 |ξ(t)| 2 ≤ σ A M 2 c 2 |ξ(t • )| 2 , ∀ t ≥ t • , (28)
while the initial conditions satisfy

W (ξ(t • )) ≤ σ A M 2 |ξ(t • )| 2 . (29)
The directional derivative of W along the trajectories of [START_REF] Ren | Consensus seeking in multiagent systems under dynamically changing interaction topologies[END_REF]

yields Ẇ (ξ(t)) ≤ ẽ⊤ [I N ⊗ AM -1 A ⊤ M ]ẽ + δ⊤ [I N ⊗ (A ⊤ A + A ⊤ BF + A ⊤ M -1 A ⊤ M )]ẽ -δ⊤ [I N ⊗ A ⊤ A] δ. ( 30
)
Then, to compact the notation, we introduce

A := A ⊤ A -A ⊤ BB ⊤ M + A ⊤ M -1 A ⊤ M
and, since A, B, and M are constant, we also introduce σ A M > 0 to denote an upper bound on the induced norm of A, that is, |A| ≤ σ A M . Hence, the terms in the second line of (30) satisfy

δ⊤ [I N ⊗ A]ẽ ≤ (σ A M ) 2 2µ |ẽ| 2 + µ 2 | δ| 2 (31)
for any µ > 0. For further development, we also introduce

γ ′ > 0 such that |AM -1 A ⊤ M | ≤ γ ′ and a ′ m > 0 such that σ A ⊤ A m ≥ a ′
m ; the latter holds under Assumption (A2).

Next, replacing (31) and

|AM -1 A ⊤ M | ≤ γ ′ in (30) yields, along trajectories, Ẇ (ξ(t)) ≤ γ ′ + (σ A M ) 2 2µ |ẽ(t)| 2 -a ′ m - µ 2 | δ(t)| 2 .
Then, we set µ = a ′ m , we define

γ ′′ := γ ′ + (σ A M ) 2
2µ , and we integrate on both sides of the latter inequality and rearrange terms to obtain

a ′ m 2 t t• | δ(τ )| 2 dτ ≤ |W (ξ(t))| + |W (ξ(t • ))| + γ ′′ t t• |ẽ(τ )| 2 dτ.
(32) Thus, using ( 28), (29), and ( 26), we obtain

t t•
Proposition 7 (Estimators' consensus) Consider the estimators defined in [START_REF] Ioannou | Robust adaptive control[END_REF], with a ij ≥ 0 such that the corresponding Laplacian matrix, whose elements are defined in (4), contains a directed spanning tree, and with ϵ ≥ σ P L M where P L solves (5) for Q L = I N , the feedback matrix F = -B ⊤ M where M solves [START_REF] Meng | Adaptive consensus for heterogeneous multi-agent systems under sensor and actuator attacks[END_REF]. Then, on the manifold {i ≤ N : (ẽ i , δi ) = (0, 0)}, the estimators (6) achieve dynamic consensus, that is, (7) holds. More precisely,

lim t→∞ |x i (t) -xm (t)| = 0, ∀ i ∈ {1, 2, • • • , N },
where xm corresponds to the solutions of ẋm = Ax m with xm (0

) := (v ⊤ l ⊗ I n )x(0). □ Proof.
In compact multi-variable form, the equations ( 6) are written as

ẋ = [I N ⊗ A]x -ϵ[L ⊗ BB ⊤ M ]x -ϵ[L ⊗ BB ⊤ M ] δ, ( 34
)
and, on the manifold {(ẽ, δ) = (0, 0)}, in view of (10), we have

ẋ = [I N ⊗ A]x -ϵ[L ⊗ BB ⊤ M ]x (35)
Now, following [START_REF] Panteley | Synchronization and dynamic consensus of heterogeneous networked systems[END_REF], to assess that the estimators (35) reach dynamic consensus, we verify that xi , for each i ≤ N , tends asymptotically to the trajectories of a weighted-average system with state xm := (v ⊤ l ⊗ I n )x. To that end, we also define the synchronization errors s xi := xi -xm . In multi-variable form, we have xm

s x = (v ⊤ l ⊗ I n )x (I N -1 N v ⊤ l ) ⊗ I n x , ( 36
)
where

s x = s ⊤ x1 • • • s ⊤ xN ⊤
. Then, differentiating on both sides of (36) and using (35), we obtain the system

ẋm ṡx = Ax m (I N ⊗ A) -ϵ(L ⊗ BB ⊤ M ) s x . ( 37
)
The latter is a dichotomous representation of the multi-agent estimators (35). On one hand, one has the decoupled average dynamics ẋm = Ax m and, on the other, the synchronization error dynamics. Therefore, establishing that s x → 0 implies that all estimators behave as the averaged model ẋm = Ax m .

Remark 8 In an ordinary consensus problem in which all systems attain a common equilibrium point, x m , such that ẋm ≡ 0. On the other hand, replacement of suitable control input can ensure that the averaged model of the multi-agent system is ẋm = Ax m , whence the terminology dynamic consensus.

• For assessing convergence of individual reference systems to the concerned averaged model, we use procedure outlined in [START_REF] Dutta | Strict Lyapunov functions for dynamic consensus in linear systems interconnected over directed graphs[END_REF]. To that end, we consider the Lyapunov function candidate

V (s x) = s ⊤ x (P L ⊗ M )s x, (38)
where P L is generated by Lemma 2 with Q L = I N . This function is positive definite under Assumption (A5) and Lemma 2.

The time derivative of V (s x), along the trajectories of (37) yields

V (s x) = s ⊤ x P L ⊗ (M A + A ⊤ M ) -ϵ(P L L + L ⊤ P L ) ⊗ (M BB ⊤ M ) s x. (39)
Then, we use ( 5), with Q L = I N , in (39), we apply the property

α[P L 1 N v ⊤ l ⊗ I n ]s x = α[P L 1 N v ⊤ l ⊗ I n ]× [(I N -1 N v ⊤ l ) ⊗ I n ]x = 0 N n
, which holds under Assumption (A4)-see Lemma 1, and the orthogonal decomposition of P L = P ⊤ L > 0, P L = T ΛT ⊤ where T ∈ R N ×N is an orthogonal matrix. Thus, (39) becomes

V (s x) = σ P L M s ⊤ x T T ⊤ ⊗ M A+A ⊤ M - ϵ σ P L M M BB ⊤ M s x,
so for any ϵ ≥ σ P L M , as posed in the Proposition 4, we have

V (s x) ≤ σ P L M s ⊤ x I N ⊗ (M A + A ⊤ M -M BB ⊤ M ) s
x and, after the algebraic Riccati equation ( 14), we get

V (s x) ≤ -σ P L M s ⊤ x (I N ⊗ Q)s
x, so global exponential stability of {s x = 0} follows invoking standard Lyapunov theory.

Practical dynamic consensus under time-varying measurement bias

To take into account the time-varying unknown bounded biases

δ i : R ≥0 → R n , i = {1, • • • , N }, the corresponding dynamics of the bias estimates is now modified as δi = -(M -1 A ⊤ M )ẽ i -βM -1 δi , (40)
where β ∈ R + . Then, we have the following. Proposition 9 (robust dynamic consensus) Consider N linear systems defined by (1) under the conditions laid in Proposition 4 and the bias update law (40) in place of [START_REF] Patre | Accommodating sensor bias in MRAC for state tracking[END_REF]. Then, the tracking and estimation errors (ẽ, δ) are uniformly ultimately bounded. Consequently, so are the tracking errors x e = x i -xi , as well as the consensus errors for the reference models, xi -xj , and for the original systems (1), x i -x j , for all i ̸ = j,.

Proof. The closed-loop equations, now using the update law (40), correspond to ė =

I N ⊗ (A + BF ) ẽ -[I N ⊗ A] δ + δ +β(I N ⊗ M -1 ) δ + I N ⊗ (M -1 A ⊤ M ) ẽ (41a) δ = I N ⊗ (M -1 A ⊤ M ) ẽ + δ + β(I N ⊗ M -1 ) δ. (41b)
Reconsider the Lyapunov function V in [START_REF] Ren | Distributed consensus in multivehicle cooperative control[END_REF]. After [START_REF] Sinha | Consensus of networked double integrator systems under sensor bias[END_REF], its total derivative along the solutions of (41a)-

(41b) yields V = -2ẽ ⊤ [I N ⊗ Q]ẽ + 2( δ + ẽ) ⊤ (I N ⊗ M ) δ + 2β( δ + ẽ) ⊤ δ -2βẽ ⊤ δ -2β| δ| 2 . (42) Therefore, V ≤ -[ 2σ Q m -β ]|ẽ| 2 -β| δ| 2 + k[ |ẽ| + | δ| ( 43
)
where

k := 2(σ M M + β)δ M ,
:= M BB ⊤ M (M A + A ⊤ M ) -1 M BB ⊤ M. Note that (M A + A ⊤ M
) is non-singular under Assumption (A2). Global ultimate boundedness of x e follows from the ultimate boundedness of δ, after integrating on both sides of (45).

Finally, to verify that the consensus error amongst the reference systems is also UUB, we reconsider the synchronization error s x and the mean field value xm of the reference systems, as defined in (36). The corresponding dynamics, in this case, is

ẋm = Ax m (46a) ṡx = (I N ⊗ A) -ϵ(L ⊗ BB ⊤ M ) s x -ϵ[L ⊗ BB ⊤ M )][(I N -1 N v ⊤ l ) ⊗ I n ] δ. ( 46b
)
Global uniform ultimate boundedness follows from the fact that the origin for (46b) with δ = 0 is globally exponentially stable and δ(t) is globally uniformly ultimately bounded.

■ 4 Simulation Results

We provide some numerical simulation results, generated using Matlab R2021a. As case-study, we consider the dynamic consensus control of five harmonic oscillators, that is, linear systems modeled as in [START_REF] Balaban | Modeling, detection, and disambiguation of sensor faults for aerospace applications[END_REF] with

A = 0 1 -1 0 , B = 0 1 , (47)
so Assumptions (A1) and (A2) hold. The oscillators are taken to form the graph showed in Figure 1, so Assumption (A5) also holds. We present the results for two tests, one in which the biases are piecewise-constant and one in which they are periodic functions. Irrespective of the nature of the measurement biases, constant or otherwise, the initial conditions assigned for the five agents are set to In the first numerical test, the biases are modeled as piecewise-constant functions taking random values within [-1, 1] and changing at random instants separated by no less than 10s; see Figure 2

L =        1 0 0 0 -1 -1 1 0 0 0 -1 0 1 0 0 0 0 -1 1 0 0 0 -1 0 1       
x 1 (0) = 3 -1 -2 1.5 2 ⊤ , x1 (0) = -1.5 1 -1.75 -0.5 2.75 ⊤ , x 2 (0) = 2 -2 -3 3 2.5 ⊤ , x2 (
The systems' trajectories are showed in Figure 3-it is noted that they all achieve dynamic consensus, as they enter in synchrony with the average oscillator. The controlled system is robust to the sudden changes in the biases. For completeness, we show the bias estimation errors in Figure 4 below. With a second numerical test, we illustrate the robustness of our consensus controller vis a vis of timevarying biases. These are of the form δ i (t) := c i + δ ′ i (t), where c i ∈ R and δ ′ i (t) is a sinusoidal or cosinusoidal signal whose amplitude and frequency is varied over time using the rand( • ) function of Matlab, so Assumption (A4) holds. See Figure 5. We set Q = 3 I 2 and β = 1.7 so the factor of ẽi in (43) be positive and we solve [START_REF] Meng | Adaptive consensus for heterogeneous multi-agent systems under sensor and actuator attacks[END_REF] for M with this new value. This yields 4.4721 1 1

2.2361

.

As per Proposition 9, the bias estimation errors are uniformly ultimately bounded-see Figure 6. Consequently, the multi-agent system achieves practical dynamic consensus-see Figure 7.

Conclusion

In a network containing a directed spanning tree, agents with linear homogeneous stabilizable dynamics can reach dynamic consensus, relying on a distributedestimation algorithm that, on one hand, consists in coupling the oscillators to the estimators and, on the other, aims at making the estimators achieve dynamic consensus themselves. The approach relies on the ability of estimating the measurement bias exactly. The latter, in turn, is guaranteed only in the case that the said bias is constant. If time-varying, ultimate boundedness is still guaranteed. That is, the original controller renders the system robust.

Current research is aimed at extending the framework to the case of heterogeneous systems; to investigate the possibility of compensating for the effects of heterogeneity via adaptive control and estimation laws. Ensuring asymptotic dynamic consensus in the presence of timevarying bias requires further study, notably, regarding a functional realistic model of the bias dynamics.

■

  with δ M := max ess sup t≥0 |δ(t)|, ess sup t≥0 | δ(t)| . Global ultimate boundedness of ẽ and δ follow from standard Lyapunov theory [7, p. 169]. To assess ultimate boundedness of the tracking errors, x e = x -x, we analyze the corresponding dynamics equation ẋe = [I N ⊗ (A + BF )]x e + (I N ⊗ BF ) δ.(44) To that end, we evaluate the total derivative of the Lyapunov function V (x e ) = x ⊤ e (I N ⊗ M )x e along the trajectories of (44), to obtainV (x e ) = x ⊤ e I N ⊗ (M A + A ⊤ M -2M BB ⊤ M ) x e -2x ⊤ e (I N ⊗ (M BB ⊤ M )) δ ≤ x ⊤ e I N ⊗ (M A + A ⊤ M -2M BB ⊤ M ) ]x e + x ⊤ e (I N ⊗ (M A + A ⊤ M ))x e + δ⊤ (I N ⊗ M ) δ ≤ -2x ⊤ e (I N ⊗ Q)x e + |M || δ| 2 , (45) where we introduced M

Fig. 1 .

 1 Fig. 1. Directed connected graph and corresponding Laplacian.

  0) = 0.5 -2 1 0.75 1.5 ⊤ , and the initial conditions for the bias estimates are set to δ1 (0) = 1.5 -1.5 2 -2.5 0.5 ⊤ , δ2 (0) = 0.5 -0.5 -2 2 3 ⊤ .

  below.

Fig. 2 .

 2 Fig. 2. Plot of piecewise-constant biases, with random changes of random amplitude.

Fig. 3 .

 3 Fig. 3. Oscillators' trajectories reaching synchronization. The solid curve in cyan represents the average oscillator.

Fig. 4 .

 4 Fig. 4. Estimation errors for the piecewise constant biases for all agents.

Fig. 5 .Fig. 6 . 5 Fig. 7 .

 5657 Fig. 5. Time-varying biases present in the state measurements of the system.

Considering t• ≥ 0 is a mere convention; the result certainly holds for all t• ∈ R.

 [START_REF] Balaban | Modeling, detection, and disambiguation of sensor faults for aerospace applications[END_REF]This work was supported by the CEFIPRA under the grant number 6001-A 2 E. Panteley and A. Loría's work is also supported by the ANR (project HANDY, contract number ANR-18-CE40-0010).


Nabil Kahouadji
email: kahouadji@math.jussieu.fr

CONSTRUCTION OF LOCAL CONSERVATION LAWS BY GENERALIZED ISOMETRIC EMBEDDINGS OF VECTOR BUNDLES

Keywords: . 1991 Mathematics Subject Classification. 58A15, 37K05, 32C22 Conservation laws, Generalized isometric embeddings of vector bundles, Exterior differential systems, Cartan-Kähler theory, Conservation laws for energy-momentum tensors

This article uses Cartan-Kähler theory to construct local conservation laws from covariantly closed vector valued differential forms, objects that can be given, for example, by harmonic maps between two Riemannian manifolds. We apply the article's main result to construct conservation laws for covariant divergence free energy-momentum tensors. We also generalize the local isometric embedding of surfaces in the analytic case by applying the main result to vector bundles of rank two over any surface.

Introduction

A conservation law can be seen as a map defined on a space F (which can be for instance, a function space, a fiber bundle section space, etc.) that associates each element f of F with a vector field X on an m-dimensional Riemannian manifold M, such that if f is a solution to a given PDE on F , the vector field X has a vanishing divergence. If we denote by g the Riemannian metric on the manifold M, we can canonically associate each vector field X ∈ Γ(TM) with a differential 1-form α X := g(X, •). Since div(X) = * d * α X (or div(X)vol M = d(X vol M )), where * is the Hodge operator, vol M is the volume form on M, and X vol M is the interior product of vol M by the vector field X, the requirement div(X) = 0 may be replaced by the requirement d(X vol M ) = 0, and hence, conservation laws may also be seen as maps on F with values on differential (m -1)-forms such that solutions to PDEs are mapped to closed differential (m -1)-forms on M. More generally, we could extend the notion of conservation laws as mapping to differential p-forms (for instance, Maxwell equations in vacuum can be expressed, as it is well-known, by requiring a system of differential 2-forms to be closed). In this paper, we address the question of finding conservation laws for a class of PDE described as follows:

Question 1. Let V be an n-dimensional vector bundle over M. Let g be a metric bundle and ∇ a connection that is compatible with that metric. We then have a covariant derivative d ∇ acting on vector valued differential forms. Assume that φ is a given covariantly closed V-valued differential p-form on M, i.e., d ∇ φ = 0.

(1.1) Does there exist N ∈ N and an embedding Ψ of V into M × R N given by Ψ(x, X) = (x, Ψ x X), where Ψ x is a linear map from V x to R N such that: • Ψ is isometric, i.e, for every x ∈ M, the map Ψ x is an isometry,

• if Ψ(φ) is the image of φ by Ψ, i.e., Ψ(φ) x = Ψ x • φ x for all x ∈ M, then dΨ(φ) = 0.

(1.2)

In this problem, the equation (1.1) represents the given PDE (or a system of PDEs) and the map Ψ plays the role of a conservation law. Note that the problem is trivial when the vector bundle is a line bundle. Indeed, the only connection on a real line bundle which is compatible with the metric is the flat one.

A fundamental example is the isometric embedding of Riemannian manifolds in Euclidean spaces and is related to the above problem as follows: M is an mdimensional Riemannian manifold, V is the tangent bundle TM, the connection ∇ is the Levi-Civita connection, p = 1 and the TM-valued differential 1-form φ is the identity map on TM. Then (1.1) expresses the torsion-free condition for the connection ∇ and any solution Ψ to (1.2) provides an isometric embedding u of the Riemannian manifold M into a Euclidean space R N through du = Ψ(φ), and conversely. An answer to the local analytic isometric embeddings of Riemannian manifolds is given by the Cartan-Janet theorem, [Car27, Jan26]. Despite the fact that the Cartan-Janet result is local and the analicity hypothesis on the data may seem to be too restrictive, the Cartan-Janet theorem is important because it actualizes the embedding in an optimal dimension unlike the Nash-Moser isometric embedding which is a smooth and global result. Consequently, if the above problem has a positive answer for p = 1, the notion of isometric embeddings of Riemannian manifolds is extended to a notion of generalized isometric embeddings of vector bundles. The general problem, when p is arbitrary, can also be viewed as an embedding of covariantly closed vector valued differential p-forms.

Another example expounded in [Hél96] of such covariantly closed vector valued differential forms is given by harmonic maps between two Riemannian manifolds. Indeed, let us consider a map u defined on an m-dimensional Riemannian manifold M with values in an n-dimensional Riemannian manifold N . On the induced bundle 1 by u over M, the u * TN -valued differential (m -1)-form * du is covariantly closed if and only if the map u is harmonic, where the connection on the induced bundle is the pull back by u of the Levi-Civita connection on N . A positive answer to the above problem in this case would make it possible to construct conservation laws on M from covariantly closed vector valued differential (m -1)-forms, provided, for example, by harmonic maps. In his book [Hél96], motivated by the question of the compactness of weakly harmonic maps in Sobolev spaces in the weak topology (which is still an open question), Hélein considers harmonic maps between Riemannian manifolds and explains how conservation laws may be obtained explicitly by the Noether's theorem if the target manifold is symmetric and formulates the problem for non symmetric target manifolds.

In this article, our main result is a positive answer when p = m -1 in the analytic case. We also find, as in the Cartan-Janet theorem, the minimal required dimension that ensures the generalized isometric embedding of an arbitrary vector bundle relative to a covariantly closed vector valued differential (m -1)-form.

Theorem 1. Let V be a real analytic n-dimensional vector bundle over a real analytic m-dimensional manifold M endowed with a metric g and a connection ∇ compatible with g. Given a non-vanishing covariantly closed V-valued differential (m -1)-form φ, there exists a local isometric embedding of

V in M × R n+κ n m,m-1 over M where κ n m,m-1 (m -1)(n -1) such that the image of φ is a conservation law.
The existence result Theorem 1 can be applied to harmonic maps. We show in the last section of this paper a further application related to energy-momentum 1 or the pull back bundle.

tensors which occur e.g. in general relativity.

The strategy for proving Theorem 1 is the following: we reformulate the problem by means of an exterior differential system on a manifold that must be defined, and since all the data involved are real analytic, we use the Cartan-Kähler theory to prove the existence of integral manifolds. The problem can be represented by the following diagram that summarizes the notations

V n ? M m (d ∇ φ) p = 0 g, ∇, Ψ M m ? M m × R N n m,p dΨ(φ)= 0 Figure 1. Generalized isometric embedding
where N is an integer that have to be defined in terms of the problem's data: n, m and p. Let us then set up a general strategy as an attempt to solve the general problem. We denote by κ n m,p the embedding codimension, i.e., the dimension of the fiber extension in order to achieve the desired embedding. Since the Cartan-Kähler theory plays an important role in this paper and since the reader may not be familiar with exterior differential systems (EDS) and the Cartan-Kähler theorem, generalities are expounded in section 2 concerning these notions and results. For details and proofs, the reader may consult Élie Cartan's book [START_REF]Les systèmes différentiels extérieurs et leurs applications géométriques[END_REF] and the third chapter of [BCG + 91].

Let us first recast our problem by using moving frames and coframes. For convenience, we adopt the following conventions for the indices: i, j, k = 1, . . . , n are the fiber indices , λ, µ, ν = 1, . . . , m are the manifold indices and a, b, c = n + 1, . . . , n + κ n m,p are the extension indices. We also adopt the Einstein summation convention, i.e., assume a summation when the same index is repeated in high and low positions. However, we will write the sign and make explicit the values of the summation indices when necessary. Let η = (η 1 , . . . , η m ) be a moving coframe on M. Let E = (E 1 , . . . , E n ) be an orthonormal moving frame of V. The covariantly closed V-valued differential p-form φ ∈ Γ(∧ p M ⊗ V n ) can be expressed as follows:

φ = E i φ i = E i ψ i λ1,...,λp η λ1,...,λp (1.3)
where ψ i λ1,...,λp are functions on M. We assume that 1 λ 1 < • • • < λ p m in the summation, and that η λ1,...,λp means

η λ1 ∧ • • • ∧ η λp . Definition 1. Let φ ∈ Γ(∧ p M ⊗ V) be a V-valued differential p-form on M.
The generalized torsion of a connection relative to φ (or for short, a φ-torsion) on a vector bundle over M is a V-valued differential (p + 1)-form Θ = (Θ i ) := d ∇ φ, i.e., in a local frame

Θ = E i Θ i := E i (dφ i + η i j ∧ φ j ) for all i (1.4)
where (η i j ) is the connection 1-form of ∇ which is an o(n)-valued differential 1-form (since ∇ is compatible with the metric bundle).

Thus, the condition of being covariantly closed d ∇ φ = 0 is equivalent to the fact that, dφ i +η i j ∧φ j = 0 for all i = 1, . . . , n. From the above definition, the connection ∇ is φ-torsion free. We also notice that the generalized torsion defined above reduces to the standard torsion in the tangent bundle case when φ = E i ψ i λ η λ = E i η i (the functions ψ i λ = δ i λ are the Kronecker tensors), and the connection is Levi-Civita. d ∇ φ = 0 ⇐⇒ dφ i + η i j ∧ φ j = 0 for all i = 1, . . . , n.

(1.5)

Assume that the problem has a solution. We consider the flat connection 1-form ω on the Stiefel space SO(n + κ n m,p )/SO(κ n m,p ), the n-adapted frames of R (n+κ n m,p ) , i.e., the set of orthonormal families of n vectors Υ = (e 1 , . . . , e n ) of R (n+κ n m,p ) which can be completed by orthonormal κ n m,p vectors (e n+1 , . . . , e n+κ n m,p ) to obtain an orthonormal set of (n + κ n m,p ) vectors. Since we work locally, we will assume without loss of generality that we are given a cross-section (e n+1 , . . . , e n+κ n m,p ) of the bundle fibration SO(n + κ n m,p ) -→ SO(n + κ n m,p )/SO(κ n m,p ). The flat standard 1-form of the connection ω is defined as follows: ω i j = e i , de j and ω a i = e a , de i , where , is the standard inner product on R n+κ n m,p . Notice that ω satisfies Cartan's structure equations. Suppose now that such an isometric embedding exists, then, if e i = Ψ(E i ), the condition dΨ(φ) = 0 yields to

e i (dφ i + ω i j ∧ φ j ) + e a (ω a i ∧ φ i ) = 0, (1.6)
a condition which is satisfied if and only if

η i j = Ψ * (ω i j ) and Ψ * (ω a i ) ∧ φ i = 0. (1.7)
The problem then turns to finding moving frames (e 1 , . . . e n , e n+1 , . . . , e n+κ n m,p ) such that there exist m-dimensional integral manifolds of the exterior ideal generated by the naive exterior differential system {ω i jη i j , ω a i ∧ φ i } on the product manifold

Σ n m,p = M × SO(n + κ n m,p ) SO(κ n m,p )
.

(1.8)

Strictly speaking, the differential forms live in different spaces. Indeed, one should consider the projections π M and π St of Σ n m,p on M and the Stiefel space and consider the ideal on Σ n m,p generated by π * M (η i j )π * St (ω i j ) and π * St (ω a i ) ∧ π * M (φ i ). It seems reasonable however to simply write {ω i jη i j , ω a i ∧ φ i }.

To find integral manifolds of the naive EDS, we would need to check that the exterior ideal is closed under the differentiation. However, this turns out not to be the case. The idea is then to add to the naive EDS the differential of the forms that generate it and therefore, we obtain a closed one.

The objects which we are dealing with in the following have a geometric meaning in the tangent bundle case with a standard 1-form (the orthonormal moving coframe, as explained above) but not in the arbitrary vector bundle case as we noticed earlier with the notion of torsion of a connection. That leads us to define notions in a generalized sense in such a way that we recover the standard notions in the tangent bundle case. First of all, the Cartan lemma, which in the isometric embedding problem implies the symmetry of the second fundamental form, does not hold. Consequently, we can not assure nor assume that the coefficients of the second fundamental form are symmetric as in the isometric embedding problem. In fact, we will show that these conditions should be replaced by generalized Cartan identities that express how coefficients of the second fundamental form are related to each other, and of course, we recover the usual symmetry in the tangent bundle case. Another difficulty is the analogue of the Bianchi identity of the curvature tensor. We will define generalized Bianchi identities relative to the covariantly closed vector valued differential p-form and a generalized curvature tensor space which corresponds, in the tangent bundle case, to the usual Bianchi identities and the Riemann curvature tensor space, respectively. Finally, besides the generalized Cartan identities and generalized curvature tensor space, we will make use of a generalized Gauss map.

The key to the proof of Theorem 1 is Lemma 1 for two main reasons: on one hand, it assures the existence of coefficients of the second fundamental form that satisfy the generalized Cartan identities and the generalized Gauss equation, properties that simplify the computation of the Cartan characters. On the other hand, the lemma gives the minimal required embedding codimension κ n m,m-1 that ensures the desired embedding. Using Lemma 1, we give another proof of Theorem 1 by an explicit construction of an ordinary integral flag. When the existence of integral manifold is established, we just need to project it on M × R n+κ n m,p .

Generalities

This section is a brief introduction to the Cartan-Kähler theory and is established to state the Cartan test, the Proposition 2 and the Cartan-Kähler theorem, results that we use in the proof of Theorem 1.

2.1. EDS and exterior ideals. Let us denote by A(M) the space of smooth differential forms on M.2 An exterior differential system is a finite set of differential forms I = {ω 1 , ω 2 , . . . , ω k } ⊂ A(M ) with which we associate the set of equations

{ω i = 0 | ω i ∈ I}. A subset of differential forms I ⊂ A(M)
is an exterior ideal if the exterior product of any differential form of I by a differential form of A(M) belongs to I and if the sum of any two differential forms of the same degree belonging to I, belongs also to I. The exterior ideal generated by I is the smallest exterior ideal containing I. An exterior ideal is said to be an exterior differential ideal if it is closed under the exterior differentiation and hence, the exterior differential ideal generated by an EDS is the smallest exterior differential ideal containing the EDS. Let us notice that an EDS is closed if and only if the exterior differential ideal and the exterior ideal generated by that EDS are equal. In particular, if I is an EDS, I ∪ dI is closed.

Introduction to Cartan-Kähler theory. Let I ⊂ A(M) be an EDS on

M and let N be a submanifold of M. The submanifold N is an integral manifold of I if ι * ϕ = 0, ∀ϕ ∈ I, where ι is an embedding ι : N -→ M. The purpose of this theory is to establish when a given EDS, which represents a PDE, has or does not have integral manifolds. We consider in this subsection, an m-dimensional real manifold M and I ⊂ A(M) an exterior differential ideal on M.

Definition 2. Let z ∈ M. A linear subspace E of T z M is an integral element of I if ϕ E =
0 for all ϕ ∈ I, where ϕ E means the evaluation of ϕ on any basis of E. We denote by V p (I) the set of p-dimensional integral elements of I.

N is an integral manifold of I if and only if each tangent space of N is an integral element of I. It is not hard to notice from the definition that a subspace of a given integral element is also an integral element. We denote by

I p = I ∩ A p (M) the set of differential p-forms of I. Thus, V p (I) = {E ∈ G p (T M) | ϕ E = 0 for all ϕ ∈ I p }.
Definition 3. Let E be an integral element of I. Let {e 1 , e 2 , . . . , e p } be a basis of E ⊂ T z M. The polar space of E, denoted by H(E), is the vector space defined as follows:

H(E) = {v ∈ T z M | ϕ(v, e 1 ,
e 2 , . . . , e p ) = 0 for all ϕ ∈ I p+1 }.

(2.9)

Notice that E ⊂ H(E). The polar space plays an important role in exterior differential system theory as we can notice from the following proposition.

Proposition 1. Let E ∈ V p (I) be a p-dimensional integral element of I. A (p + 1)- dimensional vector space E + ⊂ T z M which contains E is an integral element of I if and only if E + ⊂ H(E).
In order to check if a given p-dimensional integral element of an EDS I is contained in a (p + 1)-dimensional integral element of I , we introduce the following function called the extension rank r : V p (I) -→ Z that associates each integral element E with an integer r(E) = dimH(E) -(p + 1). The extension rank r(E) is in fact the dimension of P(H(E)/E) and is always greater than -1. If r(E) = -1, then E is contained in any (p + 1)-dimensional integral element of I and consequently, there is no hope of extending the integral element. An integral element E is said to be regular if r(E) is constant on a neighborhood of z.

Definition 4. An integral flag of I on z ∈ M of length n is a sequence (0) z ⊂ E 1 ⊂ E 2 ⊂ • • • ⊂ E n ⊂ T z M of integral elements E k of I.
An integral element E is said to be ordinary if its base point z ∈ M is an ordinary zero of I 0 = I ∩ A 0 (M) and if there exists an integral flag (0

) z ⊂ E 1 ⊂ E 2 ⊂ • • • ⊂ E n = E ⊂ T z M
where the E k , k = 1, . . . , (n -1) are regular. Moreover, if E n is itself regular, then E is said to be regular. We can now state the following important results of the Cartan-Kähler theory.

Theorem 2. (Cartan's test) Let I ⊂ A * (M) be an exterior ideal which does not contain 0-forms (functions on M). Let (0

) z ⊂ E 1 ⊂ E 2 ⊂ • • • ⊂ E n ⊂ T z M be an integral flag of I. For any k < n, we denote by C k the codimension of the polar space H(E k ) in T z M. Then V n (I) ⊂ G n (T M) is at least of codimension C 0 + C 1 + • • • + C n-1 at E n . Moreover, E n is an ordinary integral flag if and only if E n has a neighborhood U in G n (T M) such that V n (I) ∩ U is a manifold of codimension C 0 + C 1 + • • • + C n-1 in U .
The numbers C k are called Cartan characters of the k-integral element. The following proposition is useful in the applications. It allows us to compute the Cartan characters of the constructed flag in the proof of the Theorem 1.

Proposition 2. At a point z ∈ M, let E be an n-dimensional integral element of an exterior ideal I ∩ A * (M) which does not contain differential 0-forms. Let ω 1 , ω 2 , . . . , ω n , π 1 , π 2 , . . . , π s (where

s = dim M -n) be a coframe in a open neigh- borhood of z ∈ M such that E = {v ∈ T z M | π a (v) =
0 for all a = 1, . . . , s}. For all p n, we define E p = {v ∈ E | ω k (v) = 0 for all k > p}. Let {ϕ 1 , ϕ 2 , . . . , ϕ r } be the set of differential forms which generate the exterior ideal I, where ϕ ρ is of degree (d ρ + 1). For all ρ, there exists an expansion

ϕ ρ = |J|=dρ π J ρ ∧ ω J + φρ (2.10)
where the 1-forms π J ρ are linear combinations of the forms π and the terms φρ are, either of degree 2 or more on π, or vanish at z. Moreover, we have

H(E p ) = {v ∈ T z M|π J ρ (v) = 0 for all ρ and sup J p} (2.11)
In particular, for the integral flag (0

) z ⊂ E 1 ⊂ E 2 ⊂ • • • ⊂ E n ∩ T z M of I, the Cartan characters C p correspond to the number of linear independent forms {π J ρ | z such that sup J p}.
The following theorem is of great importance not only because it is a generalization of the well-known Frobenius theorem but also because it represents a generalization of the Cauchy-Kovalevskaya theorem.

Theorem 3. (Cartan-Kähler) Let I ⊂ A * (M) be a real analytic exterior differential ideal. Let X ⊂ M be a pdimensional connected real analytic Kähler-regular integral manifold of I. Suppose that r = r(X ) 0. Let Z ⊂ M be a real analytic submanifold of M of codimension r which contains X and such that T x Z and H(T x X ) are transverse in T x M for all x ∈ X ⊂ M. There exists then a (p + 1)-dimensional connected real analytic integral manifold Y of I, such that X ⊂ Y ⊂ Z. Moreover, Y is unique in the sense that another integral manifold of I having the stated properties, coincides with Y on an open neighborhood of X .

The analycity condition of the exterior differential ideal is crucial because of the requirements in the Cauchy-Kovalevskaya theorem used in the Cartan-Kähler theorem's proof. It has an important corollary. Actually, in the application, this corollary is more often used than the theorem and is sometimes called the Cartan-Kähler theorem in literature.

Corollary 1. (Cartan-Kähler)

Let I be an analytic exterior differential ideal on a manifold M. If E ⊂ T z M is an ordinary integral element of I, there exists an integral manifold of I passing through z and having E as a tangent space at point z.

One of the great applications of the Cartan-Kähler theory is the Cartan-Janet theorem concerning the local isometric embedding of Riemannian manifolds. We mention this theorem for its historical importance and because the result of this paper generalizes isometric embedding of surfaces.

Construction of Conservation Laws

In this section, we continue to explain the general strategy of solving the problem in the general case started in the introduction, and we give a complete proof of Theorem 1. In the introduction, we showed that solving the general problem is equivalent to looking for the existence of integral manifolds of the naive EDS {ω i jη i j , ω a i ∧ φ i } on the product manifold

Σ n m,p = M m × SO(n + κ n m,p ) SO(κ n m,p )
. This naive EDS is not closed. Indeed, the generalized torsion-free of the connection implies that d(ω a i ∧ φ i ) ≡ 0 modulo the naive EDS, but the Cartan's secondstructure equation yields to d(ω i j -

η i j ) ≡ a (ω a i ∧ ω a j ) -Ω i j modulo the naive EDS,
where Ω = (Ω i j ) is the curvature 2-form of the connection. Consequently, the exterior ideal that we now consider on the product manifold Σ n m,p is

I n m,p = {ω i j -η i j , a ω a i ∧ ω a j -Ω i j , ω a i ∧ φ i } alg .
(3.12)

The curvature 2-form of the connection is an o(n)-valued two form and is related to the connection 1-form (η i j ) by the Cartan's second-structure equation:

Ω i j = dη i j + η i k ∧ η k j (3.13)
A first covariant derivative of φ has led to the generalized torsion. A second covariant derivative of φ gives rise to generalized Bianchi identities 3 as follows:

d 2 ∇ (φ) = 0 ⇐⇒ Ω i j ∧ φ j = 0 for all i = 1, . . . , n. (3.14)
The conditions Ω i j ∧ φ i = 0 for all i = 1, . . . , n are called generalized Bianchi identities. We then define a generalized curvature tensor space K n m,p as the space of curvature tensor satisfying the generalized Bianchi identities:

K n m,p = {(R i j;λµ ) ∈ ∧ 2 (R n ) ⊗ ∧ 2 (R m )| Ω i j ∧ φ i = 0} (3.15)
where

Ω i j = 1 2 R i j;λµ η λ ∧ η µ = R i j;λµ η λ ⊗ η µ .
In the tangent bundle case and φ = E i η i , K n n,1 is the Riemann curvature tensor space which is of dimension 1 12 m 2 (m 2 -1).

All the data are analytic, we can apply the Cartan-Kähler theory if we are able to check the involution of the exterior differential system by constructing an m-integral flag: If the exterior ideal I n m,p passes the Cartan test, the flag is then ordinary and by the Cartan-Kähler theorem, there exist integral manifolds of I n m,p . To be able to project the product manifold Σ n m,p on M, we also need to show the existence of m-dimensional integral manifolds on which the volume form on η 1,...,m on M does not vanish.

The EDS is not involutive and hence we "prolong" it by introducing new variables. Let us express the 1-forms ω a i in the coframe (η 1 , . . . , η m ) in order to later make the computation of Cartan characters easier . Let W n m,p be an κ n m,p -dimensional Euclidean space. We then write ω a i = H a iλ η λ where H a iλ ∈ W n m,m-1 ⊗ R n ⊗ R m and define the forms π a i = ω a i -H a iλ η λ . We can also consider H iλ = (H a iλ ) as a vector of W n m,p . The forms that generate algebraically I n m,p are then expressed as follows:

a ω a i ∧ ω a j -Ω i j = a π a i ∧ π a i + a (H a jλ π a i -H a iλ π a j ) ∧ η λ + 1 2 a (H a iλ H a jµ -H a iµ H a jλ -R i j;λµ ) * η λ ∧ η µ (3.16)
and

ω a i ∧ φ i = ψ i λ1...λp π a i ∧ η λ1...λp + λ = 1, . . . , m 1 µ 1 < • • • < µp m * *
H a iλ ψ i µ1,...,µp η λµ1...µp .

(3.17)

These new expressions of the forms in terms of vectors H and the differential 1-form π will help us compute the Cartan characters of an m-integral flag. To simplify these calculations, we will choose H a iλ , which are the coefficients of the second fundamental form, so that the quantities marked with ( * ) and ( * * ) in the equations (3.16) and (3.17) vanish, and hence:

a (H a iλ H a jµ -H a iµ H a jλ ) = R i j;λµ generalized Gauss equation (3.18) λ = 1, . . . , m 1 µ 1 < • • • < µp m
H a iλ ψ i µ1,...,µp η λµ1...µp = 0 generalized Cartan identities. (3.19)

As we mentioned in the introduction, the system of equations (3.19) is said to be generalized Cartan identities because it gives us relations between the coefficients of 3 In the tangent bundle case and φ = E i η i , we recover the standard Bianchi identities of the Riemann curvature tensor, i.e,

R i jkl = R k lij and R i jkl + R i ljk + R i klj = 0 .
the second fundamental form which are not necessarily the usual symmetry given by the Cartan lemma. These properties of the coefficients and the fact that the curvature tensor (R i j;λµ ) satisfies generalized Bianchi identities yield us to name the equation (3.18) as the generalized Gauss equation.

We now define a generalized Gauss map G n m,p :

W n m,p ⊗R n ⊗R m -→ K n m,p defined for H a iλ ∈ W n m,p ⊗ R n ⊗ R m by G n m,p (H) i j;λµ = a (H a iλ H a jµ -H a iµ H a jλ ). (3.20)
Let us specialize in the conservation laws case, i.e., when p = m -1. We adopt the following notations: Λ = (1, 2, . . . , m) and Λ k = (1, . . . , k -1, k + 1, . . . , m). We thus have

η Λ = η 1 ∧ • • • ∧ η m and η Λ k = η 1 ∧ • • • ∧ η k-1 ∧ η k+1 • • • ∧ η m .
Let us construct an ordinary m-dimensional integral element of the exterior ideal I n m,m-1 on Σ n m,m-1 . Generalized Bianchi identities are trivial in this case and so dim

K n m,m-1 = n(n -1) 2 m(m -1) 2 .
The generalized Gauss equation is

H iλ .H jµ -H iµ .H jλ = R i j;λµ
, where H iλ is viewed as a vector of the κ n m,m-1 -Euclidean space W n m,m-1 . Generalized Cartan identities are λ=1,...,m (-1) λ+1 H a iλ ψ i Λ λ = 0 for all a.

(3.21)

The following lemma, for which a proof is later given, represents the key to the proof of Theorem 1.

Lemma 1. Let κ n m,m-1 (m -1)(n -1) and W n m,m-1 be a Euclidean space of dimension κ n m,m-1 . Let H n m,m-1 ⊂ W n m,m-1 ⊗ R n ⊗ R m be
the open set consisting of those elements H = (H a iλ ) so that the vectors {H iλ |i = 1, . . . , n -1 and λ = 1, . . . , m -1} are linearly independents as elements of W n m,m-1 and satisfy generalized Cartan identities. Then G n m,m-1 : ideal I n m,m-1 are as follows:

H n m,m-1 -→ K n m,m-1 is a surjective submersion. Let Z n m,m-1 = {(M, Υ, H) ∈ Σ n m,m-1 × W n m,m-1 ⊗ R n ⊗ R m | H ∈ H n m,m-1 }. We conclude from Lemma 1 that Z n m,m-1 is a submanifold 4 and hence, dim Z n m,m-1 = dim Σ n m,m-1 + dim H n m,m-1 (3.22) where dim Σ n m,m-1 = m + n(n -1) 2 + nκ n m,m-1 (3.23) dim H n m,m-1 = (nm -1)κ n m,m-1 - n(n -1)m(m -1) 4 . (3.24) We define the map Φ n m,m-1 : Z n m,m-1 -→ V m (I n m,m-1 , η Λ ) which associates (x, Υ, H) ∈ Z n m,
a ω a i ∧ ω a j -Ω i j = a π a i ∧ π a i + a (H a jλ π a i -H a iλ π a j ) ∧ η λ (3.25) ω a i ∧ φ i = ψ i λ1...λp π a i ∧ η λ1...λp . (3.26)
We recall that Cartan characters are the codimension of the polar space of integral elements. Their computations are a straightforward application of Proposition 2. and yield

C λ = n(n -1) 2 (λ + 1) for λ = 0, ..m -2 (3.27) C m-1 = n(n -1) 2 m + κ n m,m-1 (3.28) so C 0 + • • • + C m-1 = m n(n -1) 2 + n(n -1)m(m -1) 4 + κ n m,m-1 . (3.29)
Finally, the codimension of the space on m-integral elements of I n m,m-1 on which η Λ does not vanish is:

codim V m (I n m,m-1 , η Λ ) = dim G m T (x,Υ) Σ n m,m-1 -Φ(Z n m,m-1 ) = m n(n -1) 2 + n(n -1)m(m -1) 4 + κ n m,m-1 .
(3.30)

By the Cartan test, we conclude that Φ(Z n m,m-1 ) contains only ordinary m-integral flags. The Cartan-Kähler theorem then assures the existence of an m-integral manifold on which η Λ does not vanish since the exterior ideal is in involution. We finally project the integral manifold on M × R n+κ . Let us notice that the requirement of the non vanishing of the volume form η Λ on the integral manifold yields to project the integral manifold on M and also to view it as a graph of a function f defined on M with values in the space of n-adapted orthonormal frames of R n+κ . In the isometric embedding problem, the composition of f with the projection of the frames on the Euclidean space is by construction the isometric embedding map.

3.1. Another proof of Theorem 1. This proof is based on explicitly constructing an ordinary m-integral element, and the Cartan characters are computed by expliciting the polar space of an integral flag. As defined above, let us consider I n m,m-1 an exterior ideal on Σ n m,m-1 . Let us denote by (X λ ) the dual basis of (η λ ) and by (Y A ) the dual basis of (̟

A ) = (̟ σ( i j ) , ̟ σ( a j ) ) = (ω i j -η i j , ω a i ) where A = 1, . . . , dim Σ n m,m-1 -m and σ( i j ) = (j -i) + n(n -1) 2 - (n -i)(n -i + 1) 2 for 1 i < j n and σ( a i ) = n(n -1) 2 + (a -n -1)n + i for i = 1, . . . , n and a = n + 1, . . . , n + κ n m,m-1 .
Let us consider on the Grassmannian manifold G m (Σ n m,m-1 , η Λ ) a basis X λ defined as follows:

X λ (E) = X λ + P A λ (E)Y A A = 1, . . . , dim Σ n m,m-1 -m. (3.31)
Let (Π λ (E)) be the dual basis of (X λ (E)). In order to compute the codimension in the Grassmannian G m (T Σ n m,m-1 , η Λ ) of m-integral elements of I n m,m-1 , we pull back the forms that generate the exterior ideal. To do so, we evaluate the forms on the basis X λ (E) and hence the expression of the forms on the Grassmannian are:

(̟ σ( i j ) ) E = P σ( i j ) λ Π λ a ̟ σ( a i ) ∧ ̟ σ( a j ) -Ω i j E = a P σ( a i ) λ P σ( a j ) µ -P σ( a i ) µ P σ( a j ) λ -R i j;λµ Π λµ (̟ σ a j ∧ φ i ) E = λ (-1) λ+1 ψ i Λ λ P σ( a i ) λ Π Λ .
The number of functions that have linearly independent differentials represents the desired codimension, and hence with lemma 1:

codim V m (I n m,m-1 , η Λ ) = m n(n -1) 2 + n(n -1) 2 m(m -1) 2 + κ n m,m-1 (3.32)
Let us now construct an explicit m-integral element of I n m,m-1 . Since the exterior ideal does not contain any functions, every point of

Σ n m,m-1 is a 0-integral element. Let (E 0 ) z = z ∈ Σ n m,m-1 . A vector ξ in the tangent space of Σ n m,m-1 is of the form ξ = ξ λ M X λ + ξ A Y A . (3.33)
By considering the polar space, we obtain Cartan characters as previously. We then choose the integral element in the following way:

e λ = X λ + H a iλ Y σ( a i ) , (3.34)
where the coefficients H a iλ are provided by the lemma 1, which assures the existence of solutions to the successive polar systems during the construction of the integral flag. The coefficients ξ σ( i j ) for all 1 i < j n vanish for all the vectors e because of ̟ σ( i j ) . Let us denote E λ = span{e 1 , . . . , e λ }. The integral flag is then

F = E 0 ⊂ E 1 ⊂ • • • ⊂ E m-1 ⊂ E m .
Cartan characters are the same as computed previously and the Cartan test assures that the flag is ordinary. By construction, the flag does not annihilate the volume form η Λ .

Proof of lemma 1. The generalized Gauss map

G n m,m-1 defined on W n m,m-1 ⊗ R n ⊗ R m with values in K n m,m-1 is a submersion if and only if the diffe- rential dG n m,m-1 ∈ L(W n m,m-1 ⊗ R n ⊗ R m ; K n m,m-1 ), which has m(m -1)n(n -1)/4 lines and κ n m,m-1 × m × n columns, is of maximal rank.
In what follows, we make the assumption that ψ 1 Λ m = 1 and

ψ 2 Λ m = • • • = ψ n Λ m = 0.
It is always possible by changing the notation and reindexing. With this assumption, the generalized Cartan identity shows that for all a, the coefficient H a 1m on a given point of the manifold, is a linear combination of the H iλ where λ = m. When n = m = 2, we assume that the determinant detψ = (ψ 1 1 ψ 2 2ψ 1 2 ψ 2 1 ) = 0. In order to understand the proof of the submersitivity of G n m,m-1 , we first explain and show the proof for two special cases: when the vector bundle is of rank 3 over a manifold of dimension 2, and when the rank of the vector bundle is arbitrary (n 2) over a manifold of dimension 2. The proof of the surjectivity of the generalized Gauss map is established afterwards.

3.2.1. Submersitivity of the generalized Gauss map. We will proceed step by step in order to expound the proof of Lemma 1: For a warm-up, we start with the case (V 3 , M 2 , g, ∇, φ) 1 , then the case of a general vector bundle over a surface, i.e., (V n , M 2 , g, ∇) 1 , next, the case of a vector bundle of rank 2 over an m-dimensional manifold, i.e., (V 2 , M m , g, ∇, φ) m-1 , and finally, we expound the conservation laws case, i.e., (V n , M m , g, ∇, φ) m-1 .

Recall that the generalized Gauss map associates

H = (H a iλ ) with (G n m,m-1 ) i j;λµ = (H iλ H jµ -H iµ H jλ ) i j;λµ . The differential of G n m,m-1 is then: dG n m,m-1 = ∂G n m,m-1 ∂H a iλ dH a iλ (3.35) where d(G n m,m-1 ) i j;λµ = H jµ dH iλ + H iλ dH jµ -H jλ dH iµ -H iµ dH jλ . (3.36)
Denote by ǫ i j;λµ the natural basis on

K n m,m-1 = ∧ 2 R n ⊗ ∧ 2 R m . The case (V 3 , M 2 , g, ∇, φ) 1 .
Consider a vector bundle V 3 of rank 3 over a 2dimensional differentiable manifold M 2 , endowed with a metric g and a connection ∇ compatible with g. Let φ be a non-vanishing covariantly closed V 2 -valued differential 1-form. By assumption,

φ = E i φ = E i ψ i λ η λ =   1 ψ 1 2 0 ψ 2 2 0 ψ 3 2   ∧ η 1 η 2 . (3.37)
The generalized Cartan identities for each normal direction a are:

H a 12 = ψ 1 2 H a 11 + ψ 2 2 H a 21 + ψ 3 2 H a 31 . (3.38)
The curvature tensors' space is

K 3 2,1 = ∧ 2 R 3 ⊗ ∧ 2 R 2 = ∧ 2 R 3 ⊗ R = span{ǫ 1 2;12 ,
Taken into consideration the generalized Cartan identities, the differential of the generalized Gauss map is:

dG 3 2,1 =   d(G 3 2,1 ) 1 2,12 d(G 3 2,1 ) 1 3,12 d(G 3 2,1 ) 2 3,12   =   H 22 -ψ i 2 H i1 0 H 11 0 H 32 0 -ψ i 2 H i1 0 H 11 0 H 32 -H 22 -H 31 H 21   .       dH 11 dH 21 dH 31 dH 22 dH 32      
Note that H iλ are vectors in the Euclidean space W 3 2,1 of dimension κ 3 2,1 which must be determined. We want to extract from the W 3 2,1 -valued matrix dG 3 2 a submatrix of maximal rank (rank 3). Denote by L the subspace of cotangent of W 3 2,1 R 3 ⊗R 2 defined by dH 11 = dH 21 = dH 31 = 0. Then 5 dG 3 2,1 | L is : Before investigating the submersitivity of the genralized Gauss map, let us first define a flag of the subspaces of K n m,m-1 . Flag of K n m,m-1 : Let us define the following subspaces of K n m,m-1 as follows: for k = 2, . . . , n

dG 3 2 | L =  
E k | n m,m-1 = {(R i j;λµ ) ∈ K n m,m-1 |R i j;λµ = 0, if 1 i < j k and ∀1 λ < µ m} and for ν = 2, . . . , m E ν | n m,m-1 = {(R i j;λµ ) ∈ K n m,m-1 |R i j;λµ = 0, if 1 λ < µ ν and ∀1 i < j n}. By convention, E 1 | n m,m-1 = E 1 | n m,m-1 = K n m,m-1 . Therefore, 0 = E n | n m,m-1 ⊂ E n-1 | n m,m-1 ⊂ E n-2 | n m,m-1 ⊂ • • • ⊂ E 2 | n m,m-1 ⊂ K n m,m-1 0 = E m | n m,m-1 ⊂ E m-1 | n m,m-1 ⊂ E m-2 | n m,m-1 ⊂ • • • ⊂ E 2 | n m,m-1 ⊂ K n m,m-1 . Example 1 ((V 3 , M 4 , g, ∇, φ) 3 ). An element in K 3 4,3 = ∧ 2 R 3 ⊗ ∧ 2 R 4 ≃ R 18 is: R =   R 1 2;12 R 1 2;13
R 1 2;23 R 1 2;14 R 1 2;24 R 1 2;34 The case (V n , M 2 , g, ∇, φ) 1 . Recall that K n 2,1 = ∧ 2 R n ⊗ R. Some columns in the Jacobian of G n 2,1 are expressed as follows: for k = 2, . . . , n,

R 1 3;12 R 1 3;13 R 1 3;23 R 1 3;14 R 1 3;24 R 1 3;34 R 2 3;12 R 2 3;13 R 2 3;23 R 2 3;14 R 2 3;24 R 2 3;34   (3.42) and if R is in E 2 | 3 4,
dG n 2,1 ∂ ∂H a k2 = k-1 i=1 H a i1 ǫ i k;12 + (terms in E k | n 2,1 ) ∈ E k-1 | n 2,1 . (3.44)
Note that E n | n 2,1 = 0, and hence, )) a is rectangular, i.e., n(n -1)/2 lines and (κ n 2,1 × (n -1)) columns. There are actually (n -1) terms in the "diagonal" and they all have the same number of columns κ n 2,1 . The first term of the "diagonal" has one line and obviously starts at the first line, the second term has 2 lines and is at the second line, the third term has 3 lines and starts at the line number 1+2 = 3, . . . , and the last term has (n -1) lines and starts at the line number (n -2)(n -1)/2. From (3.44) and (3.45), the "diagonal" of (dG dim(E n-1 | n 2,1 ) assures that the last terms of the "diagonal", (dG n 2,1 (∂/∂H a n2 )) a , are of maximal rank.

dG n 2,1 (∂/∂H a n2 ) = n-1 i=1 H a i1 ǫ i n;12 ∈ E n-1 | n 2,1 . ( 3
⊂ E n-1 ⊂ E n-2 ⊂ • • • ⊂ E 2 ⊂ E 1 = K n 2,
The case (V n , M m , g, ∇, φ) m-1 :. For the conservation laws case, we define the following subspaces of K n m,m-1 : for k = 2, . . . , n and for ν = 2, . . . , m,

E k ν | n m,m-1 = {(R) i j;λµ ∈ K n m,m-1 |R i j;λµ = 0, if 1 i < j k and 1 λ < µ ν} and hence, E n ν | n m,m-1 = E ν | n m,m-1 and E k m | n m,m-1 = E k | n m,m-1 . By convention, E 1 ν | n m,m-1 = K n m,m-1 and E k 1 | n m,m-1 = K| n m,m-1 .
Remark 1. Let us fix ν and k. We have the same kind of flags as in (3.42) and (3.42):

E ν | n m,m-1 ⊂ E n-1 ν | n m,m-1 ⊂ E n-2 ν | n m,m-1 ⊂ • • • ⊂ E 2 ν | n m,m-1 ⊂ K n m,m-1 E k | n m,m-1 ⊂ E k m-1 | n m,m-1 ⊂ E k m-2 | n m,m-1 ⊂ • • • ⊂ E k 2 | n m,m-1 ⊂ K n m,m-1 . Example 2 ((V 3 , M 4 , g, ∇, φ) 3 -Continued). E 2 4 | 3 4,3 = E 2 | 3 4,3 , E 3 2 | 3 4,3 = E 2 | 3 4,3 ,E 3 3 | 3 4,3 = E 3 | 3 4,3 and E 3 4 | 3 4,3 = 0 and if R is in E 2 2 | 3 4,3 , E 2 3 | 3 4,
ν | n m,m-1 ⊂ E (ν-1) | n m,m-1 , for ν = 2, . . . , m, by E ν ⊂ E (ν-1) ∩E n-1 ν ⊂ E (ν-1) ∩E n-2 ν ⊂ • • • ⊂ E (ν-1) ∩E 3 ν ⊂ E (ν-1) ∩E 2 ν ⊂ E (ν-1) . ( 3
= E 5 ⊂ E 4 ∩E 3 5 ⊂ E 4 ∩ E 2 5 ⊂ E 4 ⊂ E 3 ∩ E 3 4 ⊂ E 3 ∩ E 2 4 ⊂ E 3 ⊂ E 2 ∩ E 3 3 ⊂ E 2 ∩ E 2 3 ⊂ E 2 ⊂ E 3 2 ⊂ E 2 2 ⊂ E 1 = K 4 5,4 .
We proceed in the same way to prove Lemma 1. The inclusion of the spaces E k ν | n m,m-1 is more complex and is given by the Proposition 3. We have, for k = 2, . . . , n and ν = 2, . . . , m

dG n m,m-1 (∂/∂H a kν ) = i = 1, . . . , k -1 λ = 1, . . . ν -1 H a iλ ǫ i k;λν + (terms in E k+1 ν-1 ) ∈ E k-1 ν-1 | n m,m-1 (3.48)
and since ) a which has n(n -1)m(m -1)/4 lines and κ n m,m-1 × (n -1)(m -1) columns. This matrix is of maximal rank if the vectors (H iλ ) i=1,...,(n-1) and λ=1,...,m-1 are linearly independent vectors of W n m,m-1 where κ n m,m-1

E n k | n m,m-1 = 0, dG n m,m-1 (∂/∂H a nm ) = i = 1, . . . , n -1 λ = 1, . . . m -1 H a iλ ǫ i n;λm ∈ E n-1 m-1 | n m,m-1 . ( 3
(n -1)(m -1). The minimal embedding codimension is given by the dimension of (E n-1 ∩ E m-1 | n m,m-1 ). Indeed, Proposition (3) shows that the submatrix is triangular by different sized blocks and that the terms above the block-diagonal are zero. There are (n -1)(m -1) terms in the "diagonal" and they have the same number of columns κ n m-1 . It is sufficient to show that there exists a pre-image of 0, i.e., vectors H iλ in W n m,m-1 , satisfying generalized Cartan identities and such that the set {H iλ } for i = 1, . . . , n -1 and λ = 1, . . . , m -1 are linearly independent vectors in W n m,m-1 . Indeed, the differential of the generalized Gauss map being surjective implies that G n m,m-1 (H n m,m-1 ) will contain a neighborhood of 0 in K n m,m-1 , and thus G n m,m-1 (H n m,m-1 ) = K n m,m-1 as G n m,m-1 (ρH) = ρ 2 G n m,m-1 (H).

The surjectivity of the generalized

We will construct a pre-image of 0 in H n m,m-1 . Recall that W n m,m-1 is of dimension κ n m,m-1 (n -1)(m -1). We can choose H iλ as follows:

{H iλ } i=1,...,n-1 and λ=1,...,m-1 is an orthonormal set of vectors in W n m,m-1 (3.51) We present here an application for our main result to covariant divergence free energy-momentum tensors.

H n1 = H n2 = • • • = H nm = 0 (3.
Proposition 4. Let (M m , g) be a m-dimensional real analytic Riemannian manifold, ∇ be the Levi-Civita connection and T be contravariant 2-tensor with a vanishing covariant divergence. There then exists a conservation law for T on M × R m+(m-1) 2 .

Proof. Let us consider a bivector T ∈ Γ(TM ⊗ TM) which is expressed in a chart by T = T λµ ξ λ ⊗ ξ µ , where (ξ 1 , . . . , ξ m ) is the dual basis of an orthonormal moving coframe (η 1 , . . . , η m ). The volume form is denoted by η Λ = η 1 ∧ • • • ∧ η m . Using the interior product, we can associate any bivector T with a TM-valued m-differential form τ defined as follows: Γ(TM ⊗ TM) -→ Γ(TM ⊗ ∧ (m-1) T * M)

T = T λµ ξ λ ⊗ ξ µ -→ τ = ξ λ ⊗ τ λ = ξ λ ⊗ T λµ (ξ µ η Λ ) .
The tangent space TM is endowed with the Levi-Civita connection ∇. Let us compute the covariant derivative of τ . Hence, for an m-dimensional Riemannian manifold M, the main result of this article assures the existence of an isometric embedding Ψ : T M -→ M × R m+(m-1) 2 such that d(Ψ(τ )) = 0 is a conservation law for a covariant divergence free energymomentum tensor.

For instance, if dimM = 4, then Ψ(τ ) is a closed differential 3-form on M with values in R 13 .

acknowledgements

The author is very grateful to Sara Carey for reading the manuscript and to Frédéric Hélein for his helpful remarks and suggestions.

Theorem 4 .

 4 (Cartan [Car27]-Janet [Jan26]) Every m-dimensional real analytic Riemannian manifold can be locally embedded isometrically in an m(m + 1) 2 -dimensional Euclidean space.

  m-1 with the m-plan on which the differential forms that generate algebraically I n m,m-1 vanish and the volume form η Λ on M does not vanish. Φ n m,m-1 is then an embedding and hence dim Φ(Z n m,m-1 ) = dim Z n m,m-1 .In what follows, we prove that in fact Φ(Z n m,m-1 ) contains only ordinary m-integral elements of I n m,m-1 . Since the coefficients H a iλ satisfy the generalized Gauss equation and generalized Cartan identities, the differential forms that generate the exterior 4 Z n m,m-1 is the fiber of R by a submersion. The surjectivity of G n m,m-1 assures the nonemptiness.

3 and in E 3

 3 and if R is in E 2 | 3 4,3 , E 3 | 3 4,3 and in E 4 | 3

Proposition 3 (

 3 Extension of (3.42)). For (V n , M m , g, ∇, φ) m-1 ,we can have a longer flag by replacing in (3.42) each inclusion of the type E

  Gauss map. It remains to show that the generalized Gauss map is surjective, namelyG n m,m-1 (H n m,m-1 ) = K n m,m-1 .(3.50)

  52)For j = 2, . . . , m,H jm = i = 1, . . . , n -1 λ = 1, . . . , m -1 A iλ j H iλ (3.53)whereA 1λ j = ψ j Λ λ and A iλ j = A jλ i .(3.54) 4. Conservation laws for covariant divergence free energy-momentum tensors

d

  ∇ τ = ξ λ ⊗ (dτ λ + η λ µ ∧ τ µ ) (4.55)On one hand, using the first Cartan equation that expresses the vanishing of the torsion of the Levi-Civita connection and the expression of the Christoffel symbols in terms of the connection 1-form, we obtaindτ λ = d T λµ (ξ µ η Λ ) = d(T λµ ) ∧ (ξ µ η Λ ) + T λµ d(ξ µ η Λ ) = ξ µ (T λµ ) + T λµ Γ ν νµ η Λ (4.56) and η λ µ ∧ τ µ = η λ µ ∧ T µν (ξ ν η Λ ) = T µν Γ λ νµ η Λ (4.57) consequently d ∇ τ = ξ λ ⊗ ξ µ (T λµ ) + T λµ Γ ν νµ + T µν Γ λ νµ η Λ . (4.58)On the other hand, a straightforward computation of the divergence of the bivector leads us to∇ µ T λµ = ξ µ (T λµ ) + T λµ Γ ν νµ + T µν Γ λ νµ for all λ = 1, . . . , m. (4.59)We conclude then that d ∇ τ = 0 ⇔ ∇ µ T λµ = 0 ∀λ = 1, . . . , m. (4.60)

  of maximal rank if H 11 and H 21 are linearly independent vectors.

			H 11 0 -H 31 H 21 0 H 11	  .	dH 22 dH 32	.		(3.40)
	Therefore, if κ 3 2,1	2, the matrix dG 3 2 | L is of maximal rank if H 11 and H 21
	are linearly independent vectors of W 3 2,1 . For instance, if κ 3 2,1 = 2, i.e., the normal
	directions are a = 4, 5, then
	dG 3 2 | L =	 	H 4 11 0 -H 4 31 -H 5 H 5 11 0 31 H 4 0 H 4 11 21	0 H 5 11 H 5 21	  .	   	dH 4 22 dH 5 22 dH 4 32 dH 5 32	    .	(3.41)
	5 dG 3 2,1 | L is the submatrix of dG 3 2,1 defined by: ((dG 3 2 (∂/∂H 22 ))a, (dG 3 2 (∂/∂H 23 ))a).

is

  . Note that t (H 11 , . . . , H k1 ) a is a matrix with k lines and κ n 2,1 columns. The condition of being linearly independent for the vector (H 11 , . . . H (n-1)1 ) assures that one can always extract, for each term of the diagonal, a submatrix of maximal rank. For instance, the "diagonal" term of dG n 2,1 (∂/∂H a 32 ) is t (H a 11 , H a 21 ), which is a 2 × κ n 2,1 matrix, and since the two vectors are linearly independent, there exists an invertible 2 × 2 submatrix. The same argument holds for each term of the "diagonal", and finally, κ n

		1 , the terms
	above this "diagonal" vanish in the matrix	(dG n 2,1 (∂/∂H a 22 )) a , . . . ,
	(dG n 2,1 (∂/∂H a n2 )) 2,1

a

  Example 3 ((V 4 , M 5 , g, ∇, φ) 4 ). We drop in this example the signs | 4 5,4 next to the subspaces E k

	.47)
	Note that we dropped | n m,m-1 for each subspace E, in the above equation, for more
	clarity.

ν | 4 5,4 . When we put (3.47) in (3.42), we obtain 0

This is a graded algebra under the wedge product. We do not use the standard notation Ω(M) to not confuse it with the curvature 2-form of a connection.