Holonomy of Sub-Riemannian Manifolds

1997 ◽  
Vol 08 (03) ◽  
pp. 317-344 ◽  
Author(s):  
Elisha Falbel ◽  
Claudio Gorodski ◽  
Michel Rumin
Keyword(s):  
Riemannian Manifold ◽  
Riemannian Manifolds ◽  
Smooth Manifold ◽  
Symmetric Spaces ◽  
Type Theorem ◽  
Central Symmetry ◽  
Contact Type ◽  
Riemannian Symmetric Spaces

A sub-Riemannian manifold is a smooth manifold which carries a distribution equipped with a metric. We study the holonomy and the horizontal holonomy (i.e. holonomy spanned by loops everywhere tangent to the distribution) of sub-Riemannian manifolds of contact type relative to an adapted connection. In particular, we obtain an Ambrose–Singer type theorem for the horizontal holonomy and we classify the holonomy irreducible sub-Riemannian symmetric spaces (i.e. homogeneous sub-Riemannian manifolds admitting an involutive isometry whose restriction to the distribution is a central symmetry).

Some Borsuk-Ulam-type theorems for maps from Riemannian manifolds into manifolds

1989 ◽  
Vol 111 (1-2) ◽  
pp. 61-67
Author(s):  
Duan Hai-bao
Keyword(s):  
Riemannian Manifold ◽  
Riemannian Manifolds ◽  
Symmetric Spaces ◽  
Topological Properties ◽  
Locally Symmetric Spaces ◽  
Cut Points ◽  
Continuous Map ◽  
Ulam Theorem

SynopsisSuppose f: M →N is a continuous map from a Riemannian manifold (M, d) into a manifold N. The main result of this paper is to give some conditions under which f identifies a pair of cut points. This result leads to generalisations of the classical Borsuk-Ulam theorem. As a consequence some topological properties of locally symmetric spaces are discovered.

A remark on the mixed scalar curvature of a manifold with two orthogonal totally umbilical distributions

2019 ◽  
Vol 19 (3) ◽  
pp. 291-296 ◽  
Author(s):  
Sergey Stepanov ◽  
Irina Tsyganok
Keyword(s):  
Riemannian Manifold ◽  
Scalar Curvature ◽  
Riemannian Manifolds ◽  
Type Theorem ◽  
Complete Riemannian Manifold ◽  
Warped Products ◽  
Liouville Type ◽  
Liouville Type Theorem ◽  
Totally Umbilical

Abstract We prove a Liouville-type theorem for two orthogonal complementary totally umbilical distributions on a complete Riemannian manifold with non-positive mixed scalar curvature. This is applied to some special types of complete doubly twisted and warped products of Riemannian manifolds.

scholarly journals A de Rham decomposition type theorem for contact sub-Riemannian manifolds

2021 ◽  
Vol 12 (1) ◽  
Author(s):  
Marek Grochowski
Keyword(s):  
Riemannian Manifold ◽  
Riemannian Manifolds ◽  
Type Theorem ◽  
Parallel Transport ◽  
Holonomy Group ◽  
Decomposition Theorem ◽  
Reeb Vector Field ◽  
Indefinite Metrics ◽  
De Rham ◽  
Riemannian Case

AbstractIn this paper we prove a result which can be regarded as a sub-Riemannian version of de Rham decomposition theorem. More precisely, suppose that (M, H, g) is a contact and oriented sub-Riemannian manifold such that the Reeb vector field $$\xi $$ ξ is an infinitesimal isometry. Under such assumptions there exists a unique metric and torsion-free connection on H. Suppose that there exists a point $$q\in M$$ q ∈ M such that the holonomy group $$\Psi (q)$$ Ψ ( q ) acts reducibly on H(q) yielding a decomposition $$H(q) = H_1(q)\oplus \cdots \oplus H_m(q)$$ H ( q ) = H 1 ( q ) ⊕ ⋯ ⊕ H m ( q ) into $$\Psi (q)$$ Ψ ( q ) -irreducible factors. Using parallel transport we obtain the decomposition $$H = H_1\oplus \cdots \oplus H_m$$ H = H 1 ⊕ ⋯ ⊕ H m of H into sub-distributions $$H_i$$ H i . Unlike the Riemannian case, the distributions $$H_i$$ H i are not integrable, however they induce integrable distributions $$\Delta _i$$ Δ i on $$M/\xi $$ M / ξ , which is locally a smooth manifold. As a result, every point in M has a neighborhood U such that $$T(U/\xi )=\Delta _1\oplus \cdots \oplus \Delta _m$$ T ( U / ξ ) = Δ 1 ⊕ ⋯ ⊕ Δ m , and the latter decomposition of $$T(U/\xi )$$ T ( U / ξ ) induces the decomposition of $$U/\xi $$ U / ξ into the product of Riemannian manifolds. One can restate this as follows: every contact sub-Riemannian manifold whose holonomy group acts reducibly has, at least locally, the structure of a fiber bundle over a product of Riemannian manifolds. We also give a version of the theorem for indefinite metrics.

scholarly journals Investigation of conformally killing vector fields on 5-dimensional 2-symmetric lorentzian manifolds

2021 ◽  
Vol 60 (1) ◽  
pp. 17-22
Author(s):  
Tatiana A. Andreeva ◽  
Dmitry N. Oskorbin ◽  
Evgeny D. Rodionov
Keyword(s):  
Riemannian Manifolds ◽  
Symmetric Spaces ◽  
Vector Fields ◽  
Killing Vector ◽  
Lorentzian Manifolds ◽  
Killing Vector Fields ◽  
Homogeneous Riemannian Manifolds ◽  
Locally Homogeneous ◽  
Riemannian Case ◽  
Riemannian Symmetric Spaces

Conformally Killing fields play an important role in the theory of Ricci solitons and also generate an important class of locally conformally homogeneous (pseudo) Riemannian manifolds. In the Riemannian case, V. V. Slavsky and E.D. Rodionov proved that such spaces are either conformally flat or conformally equivalent to locally homogeneous Riemannian manifolds. In the pseudo-Riemannian case, the question of their structure remains open. Pseudo-Riemannian symmetric spaces of order k, where k 2, play an important role in research in pseudo-Riemannian geometry. Currently, they have been investigated in cases k=2,3 by D.V. Alekseevsky, A.S. Galaev and others. For arbitrary k, non-trivial examples of such spaces are known: generalized Kachen - Wallach manifolds. In the case of small dimensions, these spaces and Killing vector fields on them were studied by D.N. Oskorbin, E.D. Rodionov, and I.V. Ernst with the helpof systems of computer mathematics. In this paper, using the Sagemath SCM, we investigate conformally Killing vector fields on five-dimensional indecomposable 2- symmetric Lorentzian manifolds, and construct an algorithm for their computation.

Connections with parallel skew-symmetric torsion on sub-Riemannian manifolds

2020 ◽  
pp. 49-57
Author(s):  
S. V. Galaev
Keyword(s):  
Riemannian Manifold ◽  
Riemannian Manifolds ◽  
Tensor Field ◽  
Sufficient Conditions ◽  
Contact Type ◽  
Metric Connection

On a sub-Riemannian manifold M of contact type, is considered an N-connection defined by the pair (, N), where is an interior metric connection, is an endomorphism of the distribution D. It is proved that there exists a unique N-connection such that its torsion is skew-symmetric as a contravariant tensor field. A construction of the endomorphism corresponding to such connection is found. The sufficient conditions for the obtained connection to be a metric connec­tion with parallel torsion are given.

scholarly journals $L^{p}$ harmonic 1-forms on conformally flat Riemannian manifolds

2021 ◽  
Vol 2021 (1) ◽  
Author(s):  
Jing Li ◽  
Shuxiang Feng ◽  
Peibiao Zhao
Keyword(s):  
Riemannian Manifold ◽  
Riemannian Manifolds ◽  
Schrödinger Operators ◽  
Schrodinger Operators ◽  
Finiteness Theorem ◽  
Conformally Flat ◽  
Locally Conformally Flat ◽  
Flat Riemannian Manifold ◽  
Ricci Form

AbstractIn this paper, we establish a finiteness theorem for $L^{p}$ L p harmonic 1-forms on a locally conformally flat Riemannian manifold under the assumptions on the Schrödinger operators involving the squared norm of the traceless Ricci form. This result can be regarded as a generalization of Han’s result on $L^{2}$ L 2 harmonic 1-forms.

Bézier Curves on Riemannian Manifolds and Lie Groups With Kinematics Applications

1994 ◽  
Author(s):  
Frank C. Park ◽  
Bahram Ravani
Keyword(s):  
Riemannian Manifold ◽  
Rigid Body ◽  
Lie Groups ◽  
Riemannian Manifolds ◽  
Group Structure ◽  
Recursive Algorithm ◽  
Bezier Curves ◽  
Bézier Curves ◽  
Spatial Displacements ◽  
Natural Way

Abstract In this article we generalize the concept of Bézier curves to curved spaces, and illustrate this generalization with an application in kinematics. We show how De Casteljau’s algorithm for constructing Bézier curves can be extended in a natural way to Riemannian manifolds. We then consider a special class of Riemannian manifold, the Lie groups. Because of their algebraic group structure Lie groups admit an elegant, efficient recursive algorithm for constructing Bézier curves. Spatial displacements of a rigid body also form a Lie group, and can therefore be interpolated (in the Bezier sense) using this recursive algorithm. We apply this algorithm to the kinematic problem of trajectory generation or motion interpolation for a moving rigid body.

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