On some sub-Riemannian objects in hypersurfaces of sub-Riemannian manifolds

Tan, Kang-Hai; Yang, Xiaoping

doi:10.1017/s0004972700034407

Cited by 10 publications

(4 citation statements)

References 23 publications

(66 reference statements)

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“…Theorem 2.1. [4,16] Given a sub-Riemannian manifold (M, V 0 , ), then there exists a unique nonholonomic connection satisfying…”

Section: Definition 23 the Torsion Tensor Of Nonhholonomic Connectimentioning

confidence: 99%

“…Theorem 2.1. [4,16] Given a sub-Riemannian manifold (M, V 0 , ), then there exists a unique nonholonomic connection satisfying [18] for details. On the other hand, K. Yano [17] posed a proof with a method of projecting the Riemannian connection onto the distribution to derive Theorem 2.1 in the case of Riemannian manifolds.…”

Section: Definition 23 the Torsion Tensor Of Nonhholonomic Connectionmentioning

confidence: 99%

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On semi-symmetric metric connection in sub-Riemannian manifold

Han¹,

Fu²,

Zhao³

2016

Tamkang J. Math.

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The authors first in this paper define a semi-symmetric metric nonholonomic connection (called in briefly a semi-sub-Riemannian connection) on sub-Riemannian manifolds, and study the relations between sub-Riemannian connections and semi-sub-Riemannian connections. An invariant under a connection transformation ∇ → D is obtained. The authors then further deduce a sufficient and necessary condition that a sub-Riemannian manifold associated with a semisub-Riemannian connection is flat, and derive that a sub-Riemannian manifold with vanishing curvature with respect to semi-sub-Riemannnian connection D is a group manifold if and only if it is of constant curvature.

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“…Theorem 2.1. [4,16] Given a sub-Riemannian manifold (M, V 0 , ), then there exists a unique nonholonomic connection satisfying…”

Section: Definition 23 the Torsion Tensor Of Nonhholonomic Connectimentioning

confidence: 99%

Section: Definition 23 the Torsion Tensor Of Nonhholonomic Connectionmentioning

confidence: 99%

On semi-symmetric metric connection in sub-Riemannian manifold

Han¹,

Fu²,

Zhao³

2016

Tamkang J. Math.

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“…In [6], Pappas described straight ruled surfaces and proved that a straight ruled surface in G is horizontally minimal. In [7][8][9], the geometric properties on hypersurfaces and Heisenberg groups were given by Barilari, Tan, and Balogh on Riemannian manifolds. In addition, Barilar also provided some examples of induced geometry on Heisenberg groups and hypersurfaces.…”

Section: Introductionmentioning

confidence: 99%

Sub-Riemannian Geometry of Curves and Surfaces in Roto-Translation Group Associated with Canonical Connection

Zhang,

Liu

2024

Mathematics

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The aim of this paper is to obtain the sub-Riemannian properties of the roto-translation group RT. At the same time, we compute the sub-Riemannian limits of Gaussian curvature associated with two kinds of canonical connections for a C2-smooth surface in the roto-translation group away from characteristic points and signed geodesic curvature associated with two kinds of canonical connections for C2-smooth curves on surfaces. Based on these results, we obtain a Gauss-Bonnet theorem in the RT.

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“…We intrinsically construct a sub-Laplacian on hypersurfaces in contact sub-Riemannian manifolds, which for surfaces in three-dimensional contact sub-Riemannian manifolds gives rise to the generator of the stochastic process obtained in [BBCH21] by means of Riemannian approximations, and we use our analysis to propose model cases for this setting. Some notions such as horizontal connectivity, horizontal connection and horizontal mean curvature on hypersurfaces in sub-Riemannian manifolds are studied by Tan and Yang [TY04].…”

Section: Introductionmentioning

confidence: 99%

Intrinsic sub-Laplacian for hypersurface in a contact sub-Riemannian manifold

Barilari¹,

Habermann²

2022

Preprint

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We construct and study the intrinsic sub-Laplacian, defined outside the set of characteristic points, for a smooth hypersurface embedded in a contact sub-Riemannian manifold. We prove that, away from characteristic points, the intrinsic sub-Laplacian arises as the limit of Laplace-Beltrami operators built by means of Riemannian approximations to the sub-Riemannian structure using the Reeb vector field. We carefully analyse three families of model cases for this setting obtained by considering canonical hypersurfaces embedded in model spaces for contact sub-Riemannian manifolds. In these model cases, we show that the intrinsic sub-Laplacian is stochastically complete and in particular, that the stochastic process induced by the intrinsic sub-Laplacian almost surely does not hit characteristic points. Contents 1. Introduction 1 2. Hypersurfaces in contact sub-Riemannian manifolds 7 3. Intrinsic sub-Laplacian as limit of Laplace-Beltrami operators 11 4. Canonical hypersurfaces in contact sub-Riemannian model spaces 15 References 23

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On some sub-Riemannian objects in hypersurfaces of sub-Riemannian manifolds

Cited by 10 publications

References 23 publications

On semi-symmetric metric connection in sub-Riemannian manifold

On semi-symmetric metric connection in sub-Riemannian manifold

Sub-Riemannian Geometry of Curves and Surfaces in Roto-Translation Group Associated with Canonical Connection

Intrinsic sub-Laplacian for hypersurface in a contact sub-Riemannian manifold

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