scholarly journals Local smooth isometric embeddings of low-dimensional Riemannian manifolds into Euclidean spaces

1989 ◽  
Vol 313 (1) ◽  
pp. 1-1 ◽  
Author(s):  
Gen Nakamura ◽  
Yoshiaki Maeda
Keyword(s):  
Riemannian Manifolds ◽  
Euclidean Spaces ◽  
Isometric Embeddings ◽  
Low Dimensional

scholarly journals Uniqueness of Curvature Measures in Pseudo-Riemannian Geometry

2021 ◽  
Author(s):  
Andreas Bernig ◽  
Dmitry Faifman ◽  
Gil Solanes
Keyword(s):  
Riemannian Manifolds ◽  
Riemannian Geometry ◽  
Riemannian Space ◽  
Space Forms ◽  
Type Formula ◽  
Isometric Embeddings ◽  
Curvature Measures ◽  
Riemannian Space Forms

AbstractThe recently introduced Lipschitz–Killing curvature measures on pseudo-Riemannian manifolds satisfy a Weyl principle, i.e. are invariant under isometric embeddings. We show that they are uniquely characterized by this property. We apply this characterization to prove a Künneth-type formula for Lipschitz–Killing curvature measures, and to classify the invariant generalized valuations and curvature measures on all isotropic pseudo-Riemannian space forms.

scholarly journals On low-dimensional Ricci limit spaces

2013 ◽  
Vol 209 ◽  
pp. 1-22 ◽  
Author(s):  
Shouhei Honda
Keyword(s):  
Lower Bound ◽  
Hausdorff Dimension ◽  
Ricci Curvature ◽  
Riemannian Manifolds ◽  
Limit Space ◽  
Low Dimensional ◽  
Complete Riemannian Manifolds

AbstractWe call a Gromov–Hausdorff limit of complete Riemannian manifolds with a lower bound of Ricci curvature a Ricci limit space. Furthermore, we prove that any Ricci limit space has integral Hausdorff dimension, provided that its Hausdorff dimension is not greater than 2. We also classify 1-dimensional Ricci limit spaces.

scholarly journals Warped Product Submanifolds in Locally Golden Riemannian Manifolds with a Slant Factor

Mathematics ◽  
2021 ◽  
Vol 9 (17) ◽  
pp. 2125
Author(s):  
Cristina E. Hretcanu ◽  
Adara M. Blaga
Keyword(s):  
Riemannian Manifold ◽  
Riemannian Manifolds ◽  
Warped Product ◽  
Euclidean Spaces ◽  
Slant Submanifolds

In the present paper, we study some properties of warped product pointwise semi-slant and hemi-slant submanifolds in Golden Riemannian manifolds, and we construct examples in Euclidean spaces. Additionally, we study some properties of proper warped product pointwise semi-slant (and, respectively, hemi-slant) submanifolds in a locally Golden Riemannian manifold.

scholarly journals A SPHERICAL MAPPING AND BORSUK CONJECTURE IN RIEMANNIAN AND NON-EUCLIDEAN SPACES

1994 ◽  
Vol 25 (2) ◽  
pp. 149-155
Author(s):  
BORIS V. DEKSTER
Keyword(s):  
Convex Body ◽  
Riemannian Manifolds ◽  
Additional Assumption ◽  
Convex Bodies ◽  
Smooth Convex ◽  
Euclidean Spaces ◽  
Smooth Convex Body ◽  
Bounded Set

We introduce an analog of the spherical mapping for convex bodies in a Riemannian $n$-manifold, and then use this construction to prove the Borsuk conjecture for some types of such bodies. The Borsuk conjecture is that each bounded set $X$ in the Euclidean $n$-space can be covered by $n +1$ sets of smaller diameter. The conjecture was disproved recently by Kahn and Kalai. However Hadwiger proved the Borsuk conjecture under the additional assumption that the set $X$ is a smooth convex body. Here we extend this result to convex bodies in Riemannian manifolds under some further restrictions.

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