Curvature tensors of singular spaces. (English) Zbl 1137.53341

Bernig, Andreas

Differ. Geom. Appl. 24, No. 2, 191-208 (2006).

Summary: A sequence of tensor-valued measures of certain singular spaces (e.g., subanalytic or convex sets) is constructed. The first three terms can be interpreted as scalar curvature, Einstein tensor and (modified) Riemann tensor. It is shown that these measures are independent of the ambient space, i.e., they are intrinsic. In contrast to this, there exists no intrinsic tensor-valued measure corresponding to the Ricci tensor.

MSC:

53C65	Integral geometry
32B20	Semi-analytic sets, subanalytic sets, and generalizations
53C21	Methods of global Riemannian geometry, including PDE methods; curvature restrictions

Keywords:

subanalytic sets; Einstein tensor; Riemann tensor

References:

[1]	Alesker, S., Continuous rotation invariant valuations on convex sets, Ann. of math., 149, 977-1005, (1999) · Zbl 0941.52002
[2]	Alesker, S., Description of continuous isometry covariant valuations on convex sets, Geom. dedicata, 74, 241-248, (1999) · Zbl 0935.52006
[3]	Bernig, A., Scalar curvature of definable Alexandrov spaces, Adv. geom., 2, 29-55, (2002) · Zbl 1027.53041
[4]	Bernig, A., Scalar curvature of definable CAT-spaces, Adv. geom., 3, 23-43, (2003) · Zbl 1028.53031
[5]	Bernig, A., Variation of curvatures of subanalytic spaces and schläfli-type formulas, Ann. global anal. geom., 24, 67-93, (2003) · Zbl 1037.32012
[6]	A. Bernig, Curvature bounds on subanalytic sets, Preprint, 2003 · Zbl 1037.32012
[7]	Bernig, A.; Bröcker, L., Courbures intrinsèques dans LES catégories analytico-géométrique, Ann. institut Fourier, 53, 1897-1924, (2003) · Zbl 1053.53053
[8]	Bröcker, L.; Kuppe, M., Integral geometry of tame sets, Geom. dedicata, 82, 285-323, (2000) · Zbl 1023.53057
[9]	Cheeger, J.; Müller, W.; Schrader, R., On the curvature of piecewise flat spaces, Comm. math. phys., 92, 405-454, (1984) · Zbl 0559.53028
[10]	Cheeger, J.; Colding, T.H., On the structure of spaces with Ricci curvature bounded below I, II, III, J. differential geom., J. differential geom., J. differential geom., 54, 37-74, (2000) · Zbl 1027.53043
[11]	D. Cohen-Steiner, J.-M. Morvan, Restricted Delaunay triangulations and normal cycle, in: Proc. 19th Annu. ACM Sympos. Comput. Geom., 2003, pp. 237-246 · Zbl 1422.65051
[12]	Federer, H., Geometric measure theory, (1968), Springer-Verlag Berlin · Zbl 0176.00801
[13]	Fu, J., Convergence of curvatures in secant approximations, J. differential geom., 37, 177-190, (1993) · Zbl 0794.53044
[14]	Fu, J., Curvature measures of subanalytic sets, Amer. J. math., 116, 819-880, (1994) · Zbl 0818.53091
[15]	Gallot, S.; Hulin, D.; Lafontaine, J., Riemannian geometry, (1993), Universitext, Springer-Verlag Berlin · Zbl 0636.53001
[16]	Hamm, H., On stratified Morse theory, Topology, 38, 427-438, (1999) · Zbl 0936.58007
[17]	J. Rataj, M. Zähle, General normal cycles and Lipschitz manifolds of bounded curvature, Preprint, 2004
[18]	Weyl, H., On the volume of tubes, Amer. J. math., 61, 461-472, (1939) · JFM 65.0796.01
[19]	Yano, K.; Ishihara, S., Tangent and cotangent bundles, (1973), Marcel Dekker New York · Zbl 0262.53024
[20]	Zähle, M., Approximation and characterization of generalised lipschitz – killing curvatures, Ann. global anal. geom., 8, 249-260, (1990) · Zbl 0718.53052