scholarly journals Totally umbilical submanifolds in irreducible symmetric spaces

1982 ◽  
Vol 32 (3) ◽  
pp. 332-338
Author(s):  
Bang-Yen Chen ◽  
Paul Verheyen
Keyword(s):  
Riemannian Manifold ◽  
Symmetric Spaces ◽  
Totally Umbilical ◽  
Totally Umbilical Submanifold ◽  
Totally Umbilical Submanifolds

AbstractA submanifold of a Riemannian manifold is called a totally umbilical submanifold if its first and second fundamental forms are proportional. In this paper we prove the following best possible result.

scholarly journals Classification of totally umbilical submanifolds in symmetric spaces

1980 ◽  
Vol 30 (2) ◽  
pp. 129-136 ◽  
Author(s):  
Bang-Yen Chen
Keyword(s):  
Symmetric Space ◽  
Fundamental Form ◽  
Symmetric Spaces ◽  
Second Fundamental Form ◽  
Totally Umbilical ◽  
The Second Fundamental Form ◽  
Totally Umbilical Submanifold ◽  
Totally Umbilical Submanifolds ◽  
First Fundamental Form

AbstractA submanifold of a Riemannian manifold is called a totally umbilical submanifold if the second fundamental form is proportional to the first fundamental form. In this paper, we shall prove that there is no totally umbilical submanifold of codimension less than rank M — 1 in any irreducible symmetric space M. Totally umbilical submanifolds of higher codimensions in a symmetric space are also studied. Some classification theorems of such submanifolds are obtained.

scholarly journals On pinching theorems for compact pseudo-umbilical submanifold

2012 ◽  
Vol 45 (3) ◽  
pp. 645-654
Author(s):  
Jing Mao ◽  
Shaodong Qin
Keyword(s):  
Riemannian Manifold ◽  
Constant Curvature ◽  
Mean Curvature ◽  
Sufficient Conditions ◽  
Curvature Vector ◽  
Mean Curvature Vector ◽  
Parallel Mean Curvature Vector ◽  
Totally Umbilical ◽  
Totally Umbilical Submanifold ◽  
Parallel Mean Curvature

AbstractConsider submanifolds in the nested space. For a compact pseudoumbilical submanifold with parallel mean curvature vector of a Riemannian submanifold with constant curvature immersed in a quasi-constant curvature Riemannian manifold, two sufficient conditions are given to let the pseudo-umbilical submanifold become a totally umbilical submanifold.

Conformal Killing Forms on Totally Umbilical Submanifolds

2016 ◽  
Vol 217 (5) ◽  
pp. 525-539 ◽  
Author(s):  
S. E. Stepanov ◽  
I. A. Alexandrova ◽  
I. I. Tsyganok ◽  
J. Mikeš
Keyword(s):  
Conformal Killing ◽  
Totally Umbilical ◽  
Totally Umbilical Submanifolds

scholarly journals Totally umbilical submanifolds of quaternion-space-forms

1978 ◽  
Vol 26 (2) ◽  
pp. 154-162 ◽  
Author(s):  
Bang-Yen Chen
Keyword(s):  
Space Forms ◽  
Totally Umbilical ◽  
Quaternion Space ◽  
Totally Umbilical Submanifolds

AbstractTotally umbilical submanifolds of dimension greater than four in quaternion-space-forms are completely classified.

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.

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).

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