Generalized Curvatures

Front Cover
Springer Science & Business Media, May 13, 2008 - Mathematics - 266 pages
The central object of this book is the measure of geometric quantities describing N a subset of the Euclidean space (E ,), endowed with its standard scalar product. Let us state precisely what we mean by a geometric quantity. Consider a subset N S of points of the N-dimensional Euclidean space E , endowed with its standard N scalar product. LetG be the group of rigid motions of E . We say that a 0 quantity Q(S) associated toS is geometric with respect toG if the corresponding 0 quantity Q[g(S)] associated to g(S) equals Q(S), for all g?G . For instance, the 0 diameter ofS and the area of the convex hull ofS are quantities geometric with respect toG . But the distance from the origin O to the closest point ofS is not, 0 since it is not invariant under translations ofS. It is important to point out that the property of being geometric depends on the chosen group. For instance, ifG is the 1 N group of projective transformations of E , then the property ofS being a circle is geometric forG but not forG , while the property of being a conic or a straight 0 1 line is geometric for bothG andG . This point of view may be generalized to any 0 1 subsetS of any vector space E endowed with a groupG acting on it.
 

Contents

Introduction
2
The Theory of Normal Cycles
8
Curves
13
Surfaces
29
4
47
Elements of Measure Theory
57
6
71
Convex Subsets
77
Currents
121
Approximation of the Volume
129
14
139
16
153
23
221
Curvature Measures in E3
231
Approximation of the Curvature of Curves
241
Approximation of the Curvatures of Surfaces
249

8
91
Background on Riemannian Geometry
101
Riemannian Submanifolds
109

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