Some Conjectures on Intrinsic Volumes of Riemannian Manifolds and Alexandrov Spaces

Some Conjectures on Intrinsic Volumes of Riemannian Manifolds and Alexandrov Spaces For any closed smooth Riemannian manifold Weyl (Am J Math 61:461–472, 1939) has defined a sequence of numbers called today intrinsic volumes. They include volume, Euler characteristic, and integral of the scalar curvature. We conjecture that absolute values of all intrinsic volumes are bounded by a constant depending only on the dimension of the manifold, upper bound on its diameter, and lower bound on the sectional curvature. Furthermore we conjecture that intrinsic volumes can be defined for some (so called weakly smoothable) Alexandrov spaces with curvature bounded below and state few of the expected properties of them, particularly the behavior under the Gromov-Hausdorff limits. We suggest conjectural compactifications of the space of smooth closed connected Riemannian manifolds with given upper bounds on dimension and diameter and a lower bound on sectional curvature to which the intrinsic volumes extend by continuity. We discuss also known cases of some of these conjectures. The work is a joint project with Petrunin. http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png Arnold Mathematical Journal Springer Journals

Some Conjectures on Intrinsic Volumes of Riemannian Manifolds and Alexandrov Spaces

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Publisher
Springer International Publishing
Copyright
Copyright © 2018 by Institute for Mathematical Sciences (IMS), Stony Brook University, NY
Subject
Mathematics; Mathematics, general
ISSN
2199-6792
eISSN
2199-6806
D.O.I.
10.1007/s40598-017-0078-6
Publisher site
See Article on Publisher Site

Abstract

For any closed smooth Riemannian manifold Weyl (Am J Math 61:461–472, 1939) has defined a sequence of numbers called today intrinsic volumes. They include volume, Euler characteristic, and integral of the scalar curvature. We conjecture that absolute values of all intrinsic volumes are bounded by a constant depending only on the dimension of the manifold, upper bound on its diameter, and lower bound on the sectional curvature. Furthermore we conjecture that intrinsic volumes can be defined for some (so called weakly smoothable) Alexandrov spaces with curvature bounded below and state few of the expected properties of them, particularly the behavior under the Gromov-Hausdorff limits. We suggest conjectural compactifications of the space of smooth closed connected Riemannian manifolds with given upper bounds on dimension and diameter and a lower bound on sectional curvature to which the intrinsic volumes extend by continuity. We discuss also known cases of some of these conjectures. The work is a joint project with Petrunin.

Journal

Arnold Mathematical JournalSpringer Journals

Published: May 22, 2018

References

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