Riemannian manifolds of conullity two admitting semiparallel isometric immersions

2004 ◽  
Vol 53 (4) ◽  
pp. 203
Author(s):  
Ü Lumiste
Keyword(s):  
Riemannian Manifolds ◽  
Isometric Immersions

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 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 Invariant submanifolds in flow geometry

1997 ◽  
Vol 62 (3) ◽  
pp. 290-314
Author(s):  
J. C. González-Dávila ◽  
M. C. González-Dávila ◽  
L. Vanhecke
Keyword(s):  
Riemannian Manifolds ◽  
Killing Vector ◽  
Killing Vector Field ◽  
Normal Flow ◽  
Isometric Immersions ◽  
Flow Lines ◽  
Space Forms ◽  
Flow Geometry ◽  
Invariant Submanifolds ◽  
Riemannian Flow

AbstractWe begin a study of invariant isometric immersions into Riemannian manifolds (M, g) equipped with a Riemannian flow generated by a unit Killing vector field ξ. We focus our attention on those (M, g) where ξ is complete and such that the reflections with respect to the flow lines are global isometries (that is, (M, g) is a Killing-transversally symmetric space) and on the subclass of normal flow space forms. General results are derived and several examples are provided.

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