TY - JOUR

T1 - Some Conjectures on Intrinsic Volumes of Riemannian Manifolds and Alexandrov Spaces

AU - Alesker, Semyon

N1 - Publisher Copyright:
© 2018, Institute for Mathematical Sciences (IMS), Stony Brook University, NY.

PY - 2018/4/1

Y1 - 2018/4/1

N2 - 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.

AB - 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.

KW - Alexandrov space

KW - Gromov-Hausdorff convergence

KW - Intrinsic volumes

KW - Riemannian manifold

UR - http://www.scopus.com/inward/record.url?scp=85048232259&partnerID=8YFLogxK

U2 - 10.1007/s40598-017-0078-6

DO - 10.1007/s40598-017-0078-6

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AN - SCOPUS:85048232259

SN - 2199-6792

VL - 4

SP - 1

EP - 17

JO - Arnold Mathematical Journal

JF - Arnold Mathematical Journal

IS - 1

ER -