Generalizations of Frobenius’ Theorem on Manifolds and Subcartesian Spaces

Jędrzej Śniatycki

doi:10.4153/cmb-2007-044-2

Generalizations of Frobenius’ Theorem on Manifolds and Subcartesian Spaces

Canadian Mathematical Bulletin ◽

10.4153/cmb-2007-044-2 ◽

2007 ◽

Vol 50 (3) ◽

pp. 447-459 ◽

Cited By ~ 1

Author(s):

Jędrzej Śniatycki

Keyword(s):

Vector Fields ◽

Integral Manifold ◽

Constant Rank ◽

Frobenius Theorem

AbstractLet be a family of vector fields on a manifold or a subcartesian space spanning a distribution D. We prove that an orbit O of is an integral manifold of D if D is involutive on O and it has constant rank on O. This result implies Frobenius’ theorem, and its various generalizations, on manifolds as well as on subcartesian spaces.

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Closed 𝓐-p Quasiconvexity and Variational Problems with Extended Real-Valued Integrands

ESAIM Control Optimisation and Calculus of Variations ◽

10.1051/cocv/2017062 ◽

2018 ◽

Vol 24 (4) ◽

pp. 1605-1624

Author(s):

Adam Prosinski

Keyword(s):

Vector Fields ◽

Additional Assumption ◽

Integral Functional ◽

Variational Problems ◽

Constant Rank ◽

Differential Constraint ◽

First Order ◽

Characteristic Cone ◽

Lower Semi ◽

Continuity Assumption

This paper relates the lower semi-continuity of an integral functional in the compensated compactness setting of vector fields satisfying a constant-rank first-order differential constraint, to closed 𝓐-p quasiconvexity of the integrand. The lower semi-continuous envelope of relaxation is identified for continuous, but potentially extended real-valued integrands. We discuss the continuity assumption and show that when it is dropped our notion of quasiconvexity is still equivalent to lower semi-continuity of the integrand under an additional assumption on the characteristic cone of 𝓐.

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Integrability of singular distributions on Banach manifolds

Mathematical Proceedings of the Cambridge Philosophical Society ◽

10.1017/s0305004100052130 ◽

1976 ◽

Vol 79 (1) ◽

pp. 117-128 ◽

Cited By ~ 4

Author(s):

David Chillingworth ◽

Peter Stefan

Keyword(s):

Integral Manifold ◽

Sufficient Conditions ◽

Infinite Dimensions ◽

Frobenius Theorem ◽

Singular Foliation ◽

Infinite Dimensional ◽

Banach Manifolds ◽

Integrable Distribution ◽

Real Analytic ◽

Necessary And Sufficient

One of the key results in the work of the second author ((7), (8)) on integrability of systems of vectorfields is the theorem which relates integrability of a distribution to the concept of homogeneity. In this paper, we show that the homogeneity theorem also applies in an infinite-dimensional context, and this allows us to derive infinite-dimensional versions of several further results in (7) and (8), formulated in terms of distributions. In particular, we are able to express necessary and sufficient conditions for homogeneity in terms of Lie brackets (Theorems 3 and 4) and to characterize integrable real-analytic distributions (Theorem 5). As a corollary to our Theorem 2, we recover the standard Frobenius theorem on the integrability of regular distributions. We also discuss briefly a basic problem which arises in infinite dimensions when we view an integral manifold of an integrable distribution as part of a singular foliation.

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Invariant submanifolds for systems of vector fields of constant rank

Science China Mathematics ◽

10.1007/s11425-016-5139-0 ◽

2016 ◽

Vol 59 (7) ◽

pp. 1417-1426

Author(s):

HeungJu Ahn ◽

ChongKyu Han

Keyword(s):

Vector Fields ◽

Constant Rank ◽

Invariant Submanifolds

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Burgers’ Equations in the Riemannian Geometry Associated with First-Order Differential Equations

Advances in Mathematical Physics ◽

10.1155/2018/7590847 ◽

2018 ◽

Vol 2018 ◽

pp. 1-8 ◽

Cited By ~ 1

Author(s):

Z. Ok Bayrakdar ◽

T. Bayrakdar

Keyword(s):

Vector Fields ◽

Integral Manifold ◽

Integrability Condition ◽

Three Dimensional ◽

Dimensional Manifold ◽

Jet Bundle ◽

First Order ◽

Burgers Equations ◽

Integral Curves ◽

Metric Connection

We construct metric connection associated with a first-order differential equation by means of the generator set of a Pfaffian system on a submanifold of an appropriate first-order jet bundle. We firstly show that the inviscid and viscous Burgers’ equations describe surfaces attached to an ODE of the form dx/dt=u(t,x) with certain Gaussian curvatures. In the case of PDEs, we show that the scalar curvature of a three-dimensional manifold encoding a system of first-order PDEs is determined in terms of the integrability condition and the Gaussian curvatures of the surfaces corresponding to the integral curves of the vector fields which are annihilated by the contact form. We see that an integral manifold of any PDE defines intrinsically flat and totally geodesic submanifold.

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