scholarly journals Regularity of isometric immersions of positively curved Riemannian manifolds and its analogy with CR geometry

1988 ◽  
Vol 28 (3) ◽  
pp. 477-484
Author(s):  
Chong-Kyu Han
Keyword(s):  
Riemannian Manifolds ◽  
Isometric Immersions ◽  
Cr Geometry ◽  
Positively Curved

SOME EXISTENCE AND NONEXISTENCE RESULTS OF ISOMETRIC IMMERSIONS OF RIEMANNIAN MANIFOLDS

2004 ◽  
Vol 06 (06) ◽  
pp. 867-879 ◽  
Author(s):  
ZIZHOU TANG
Keyword(s):  
Euclidean Space ◽  
Projective Plane ◽  
Riemannian Manifolds ◽  
Unit Sphere ◽  
Klein Bottle ◽  
Euler Class ◽  
Isometric Immersions ◽  
Existence And Nonexistence

This paper investigates existence and non-existence of immersions of Riemannian manifolds. It discovers the lowest dimension of the Euclidean space into which the projective plane FP2 is isometrically immersed, by the computation of the normal Euler class. For strictly hyperbolic immersion, a new obstruction involving signature or Kervaire semi-characteristic is found. As for the existence, it constructs a strictly hyperbolic immersion from the Klein bottle to the unit sphere S3(1), solving a question posed by Gromov.

scholarly journals Riemannian manifolds with local symmetry

2014 ◽  
Vol 06 (02) ◽  
pp. 211-236 ◽  
Author(s):  
Wouter van Limbeek
Keyword(s):  
Riemannian Manifolds ◽  
Local Symmetry ◽  
Universal Cover ◽  
Connected Components ◽  
Curved Manifolds ◽  
Locally Homogeneous ◽  
Simply Connected ◽  
Positively Curved ◽  
Aspherical Manifolds

We give a classification of many closed Riemannian manifolds M whose universal cover [Formula: see text] possesses a nontrivial amount of symmetry. More precisely, we consider closed Riemannian manifolds M such that [Formula: see text] has noncompact connected components. We prove that in many cases, such a manifold is as a fiber bundle over a locally homogeneous space. This is inspired by work of Eberlein (for non-positively curved manifolds) and Farb-Weinberger (for aspherical manifolds), and generalizes work of Frankel (for a semisimple group action). As an application, we characterize simply-connected Riemannian manifolds with both compact and finite volume noncompact quotients.

scholarly journals A Decomposition Theorem for Immersions of Product Manifolds

2015 ◽  
Vol 59 (1) ◽  
pp. 247-269 ◽  
Author(s):  
Ruy Tojeiro
Keyword(s):  
Riemannian Manifolds ◽  
Space Form ◽  
Type Theorem ◽  
Decomposition Theorem ◽  
Product Manifold ◽  
Isometric Immersions ◽  
Product Metrics ◽  
Special Cases ◽  
De Rham ◽  
Shape Operators

AbstractWe introduce polar metrics on a product manifold, which have product and warped product metrics as special cases. We prove a de Rham-type theorem characterizing Riemannian manifolds that can be locally or globally decomposed as a product manifold endowed with a polar metric. For such a product manifold, our main result gives a complete description of all its isometric immersions into a space form whose second fundamental forms are adapted to its product structure in the sense that the tangent spaces to each factor are preserved by all shape operators. This is a far-reaching generalization of a basic decomposition theorem for isometric immersions of Riemannian products due to Moore as well as of its extension by Nölker to isometric immersions of warped products.

scholarly journals Contact Semi-Riemannian Structures in CR Geometry: Some Aspects

Axioms ◽  
2019 ◽  
Vol 8 (1) ◽  
pp. 6 ◽  
Author(s):  
Domenico Perrone
Keyword(s):  
Riemannian Manifolds ◽  
Integrability Condition ◽  
Cr Manifolds ◽  
Cr Geometry ◽  
Cr Structure ◽  
Cr Structures ◽  
Similarities And Differences ◽  
One To One ◽  
Riemannian Case

There is one-to-one correspondence between contact semi-Riemannian structures ( η , ξ , φ , g ) and non-degenerate almost CR structures ( H , ϑ , J ) . In general, a non-degenerate almost CR structure is not a CR structure, that is, in general the integrability condition for H 1 , 0 : = X - i J X , X ∈ H is not satisfied. In this paper we give a survey on some known results, with the addition of some new results, on the geometry of contact semi-Riemannian manifolds, also in the context of the geometry of Levi non-degenerate almost CR manifolds of hypersurface type, emphasizing similarities and differences with respect to the Riemannian case.

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