Isometric immersions of complete Riemannian manifolds into Euclidean space

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.

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On isometric immersions of Riemannian manifolds in euclidean space

Boletim da Sociedade Brasileira de Matemática ◽

10.1007/bf02614960 ◽

1970 ◽

Vol 1 (2) ◽

pp. 31-45 ◽

Cited By ~ 9

Author(s):

R. H. Szczarba

Keyword(s):

Euclidean Space ◽

Riemannian Manifolds ◽

Isometric Immersions

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Sharp Caffarelli–Kohn–Nirenberg inequalities on Riemannian manifolds: the influence of curvature

Proceedings of the Royal Society of Edinburgh Section A Mathematics ◽

10.1017/prm.2020.100 ◽

2021 ◽

pp. 1-26

Author(s):

Van Hoang Nguyen

Keyword(s):

Euclidean Space ◽

Riemannian Manifolds ◽

Remainder Term ◽

Sharp Constant ◽

Hadamard Manifolds ◽

Best Constant ◽

Quantitative Version ◽

Rigidity Results ◽

Ckn Inequalities ◽

Complete Riemannian Manifolds

We first establish a family of sharp Caffarelli–Kohn–Nirenberg type inequalities (shortly, sharp CKN inequalities) on the Euclidean spaces and then extend them to the setting of Cartan–Hadamard manifolds with the same best constant. The quantitative version of these inequalities also is proved by adding a non-negative remainder term in terms of the sectional curvature of manifolds. We next prove several rigidity results for complete Riemannian manifolds supporting the Caffarelli–Kohn–Nirenberg type inequalities with the same sharp constant as in the Euclidean space of the same dimension. Our results illustrate the influence of curvature to the sharp CKN inequalities on the Riemannian manifolds. They extend recent results of Kristály (J. Math. Pures Appl. 119 (2018), 326–346) to a larger class of the sharp CKN inequalities.

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SMOLYANOV–WEIZSÄCKER SURFACE MEASURES GENERATED BY DIFFUSIONS ON THE SET OF TRAJECTORIES IN RIEMANNIAN MANIFOLDS

Infinite Dimensional Analysis Quantum Probability and Related Topics ◽

10.1142/s0219025708002951 ◽

2008 ◽

Vol 11 (01) ◽

pp. 21-31 ◽

Cited By ~ 6

Author(s):

ILYA V. TELYATNIKOV

Keyword(s):

Brownian Motion ◽

Euclidean Space ◽

Diffusion Process ◽

Riemannian Manifolds ◽

Marginal Distribution ◽

Diffusion Processes ◽

Ambient Space ◽

Time Interval ◽

Standard Brownian Motion ◽

Limit Surface

We consider surface measures on the set of trajectories in a smooth compact Riemannian submanifold of Euclidean space generated by diffusion processes in the ambient space. A construction of surface measures on the path space of a smooth compact Riemannian submanifold of Euclidean space was introduced by Smolyanov and Weizsäcker for the case of the standard Brownian motion. The result presented in this paper extends the result of Smolyanov and Weizsäcker to the case when we consider measures generated by diffusion processes in the ambient space with nonidentical correlation operators. For every partition of the time interval, we consider the marginal distribution of the diffusion process in the ambient space under the condition that it visits the manifold at all times of the partition, when the mesh of the partition tends to zero. We prove the existence of some limit surface measures and the equivalence of the above measures to the distribution of some diffusion process on the manifold.

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