Integral Geometry of Tame Sets

Curvature measures on certain tame Whitney-stratified sets are defined as coefficients of modified volume-growth polynomials. Stratified Morse theory yields alternative descriptions of these curvature measures for tame (possibly highly singular) sets. From this we obtain a generalized Gauss–Bonnet formula and various kinematic formulas. Finally, for O-minimal sets it is shown that curvature measures only depend on the inner metric.

[1]  Prof. Dr. W., BLASCHKE, Vorlesungen über Integralgeometrie. 1. Heft. (Hamburger math. Einzelschriften 20. Heft). 2. Aufl, 60 S. Leipzig 1936, Verlag B. C. Teubner. Preis brosch. 5 M, geb. 6 M , 1936 .

[2]  W. Blaschke.  Vorlesungen über Integralgeometrie , 1937 .

[3]  H. Weyl.  On the Volume of Tubes , 1939 .

[4]  C. B. Allendoerfer.  The Euler Number of a Riemann Manifold , 1940 .

[5]  C. B. Allendoerfer,et al.  The Gauss-Bonnet theorem for Riemannian polyhedra , 1943 .

[6]  S. Chern.  A simple instrinsic proof of the Gauss Bonnet formula for closed Riemannian manifolds , 1944 .

[7]  H. Hadwiger.  Vorlesungen über Inhalt, Oberfläche und Isoperimetrie , 1957 .

[8]  H. Whitney.  Tangents to an Analytic Variety , 1965 .

[9]  Shiing-Shen Chern,et al.  On the Kinematic Formula in Integral Geometry , 1966 .

[10]  T. Banchoff.  CRITICAL POINTS AND CURVATURE FOR EMBEDDED POLYHEDRA , 1967 .

[11]  Totale Absolutkrummung in Differentialgeometrie und -topologie , 1968 .

[12]  H. Fédérer.  Geometric Measure Theory , 1969 .

[13]  René Thom,et al.  Ensembles et morphismes stratifiés , 1969 .

[14]  R. Hardt,et al.  Topological properties of subanalytic sets , 1975 .

[15]  L. Santaló.  Integral geometry and geometric probability , 1976 .

[16]  Robert Schrader,et al.  On the curvature of piecewise flat spaces , 1984 .

[17]  Robert Schrader,et al.  Kinematic and tube formulas for piecewise linear spaces , 1986 .

[18]  Curvatures and currents for unions of sets with positive reach , 1987 .

[19]  M. Goresky,et al.  Stratified Morse theory , 1988 .

[20]  Joseph H. G. Fu.  Curvature measures and generalized Morse theory , 1989 .

[21]  J. Fu,et al.  Curvature Measures of Subanalytic Sets , 1994 .

[22]  L. Dries,et al.  Geometric categories and o-minimal structures , 1996 .

[23]  M. Shiota.  Geometry of subanalytic and semialgebraic sets , 1997 .

[24]  L. van den Dries,et al.  Tame Topology and O-minimal Structures , 1998 .

[25]  H. Hamm.  ON STRATIFIED MORSE THEORY , 1999 .