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The structure of reachable sets for affine control systems induced by generalized Martinet sub-lorentzian metrics

Marek Grochowski (2012)

ESAIM: Control, Optimisation and Calculus of Variations

In this paper we investigate analytic affine control systems q ˙ q̇ = X + uY, u ∈  [a,b] , where X,Y is an orthonormal frame for a generalized Martinet sub-Lorentzian structure of order k of Hamiltonian type. We construct normal forms for such systems and, among other things, we study the connection between the presence of the singular trajectory starting at q0 on the boundary of the reachable set from q0 with the minimal number of analytic functions needed for describing the reachable set from q0.

The Tanaka-Webster connection for almost 𝒮 -manifolds and Cartan geometry

Antonio Lotta, Anna Maria Pastore (2004)

Archivum Mathematicum

We prove that a CR-integrable almost 𝒮 -manifold admits a canonical linear connection, which is a natural generalization of the Tanaka–Webster connection of a pseudo-hermitian structure on a strongly pseudoconvex CR manifold of hypersurface type. Hence a CR-integrable almost 𝒮 -structure on a manifold is canonically interpreted as a reductive Cartan geometry, which is torsion free if and only if the almost 𝒮 -structure is normal. Contrary to the CR-codimension one case, we exhibit examples of non normal...

The works of Charles Ehresmann on connections: from Cartan connections to connections on fibre bundles

Charles-Michel Marle (2007)

Banach Center Publications

Around 1923, Élie Cartan introduced affine connections on manifolds and defined the main related concepts: torsion, curvature, holonomy groups. He discussed applications of these concepts in Classical and Relativistic Mechanics; in particular he explained how parallel transport with respect to a connection can be related to the principle of inertia in Galilean Mechanics and, more generally, can be used to model the motion of a particle in a gravitational field. In subsequent papers, Élie Cartan...

Three dimensional near-horizon metrics that are Einstein-Weyl

Matthew Randall (2017)

Archivum Mathematicum

We investigate which three dimensional near-horizon metrics g N H admit a compatible 1-form X such that ( X , [ g N H ] ) defines an Einstein-Weyl structure. We find explicit examples and see that some of the solutions give rise to Einstein-Weyl structures of dispersionless KP type and dispersionless Hirota (aka hyperCR) type.

Timelike B 2 -slant helices in Minkowski space E 1 4

Ahmad T. Ali, Rafael López (2010)

Archivum Mathematicum

We consider a unit speed timelike curve α in Minkowski 4-space 𝐄 1 4 and denote the Frenet frame of α by { 𝐓 , 𝐍 , 𝐁 1 , 𝐁 2 } . We say that α is a generalized helix if one of the unit vector fields of the Frenet frame has constant scalar product with a fixed direction U of 𝐄 1 4 . In this work we study those helices where the function 〈 𝐁 2 , U 〉 is constant and we give different characterizations of such curves.

Torsion and the second fundamental form for distributions

Geoff Prince (2016)

Communications in Mathematics

The second fundamental form of Riemannian geometry is generalised to the case of a manifold with a linear connection and an integrable distribution. This bilinear form is generally not symmetric and its skew part is the torsion. The form itself is closely related to the shape map of the connection. The codimension one case generalises the traditional shape operator of Riemannian geometry.

Totally Umbilical Pseudo-Slant Submanifolds of a Nearly Cosymplectic Manifold

Khan, M. A., Uddin, Siraj, Singh, Khushwant (2010)

Serdica Mathematical Journal

2000 Mathematics Subject Classification: 53C40, 53B25.In the present note we study totally umbilical pseudo-slant submanifolds of a nearly cosymplectic manifold. We have obtained a classification theorem for totally umbilical pseudo-slant submanifolds of a nearly cosymplectic manifold.

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