# Generalized Curvatures

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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.
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Springer Berlin Heidelberg; May 2008

274 pages; ISBN 9783540737926

Read online, or download in secure PDF format

Title: Generalized Curvatures

Author: Jean-Marie Morvan

274 pages; ISBN 9783540737926

Read online, or download in secure PDF format

Title: Generalized Curvatures

Author: Jean-Marie Morvan

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Subject categories

- Academic > Engineering > Electrical engineering. Electronics. Nuclear engineering > Electronics
- Academic > Mathematics > Geometry. Trigonometry.Topology > Geometric tomography
- Academic > Mathematics > Tables > Problem solving
- Mathematics > Number Theory
- Mathematics > Geometry
- Computers > Computer Vision
- Computers > Computer Graphics

ISBNs

3540737928

9783540737919

9783540737926

### In the press

From the reviews:

"This book is a welcome addition to the literature in differential geometry. The main aim of this book is the measure of geometric quantities describing a subset of the Euclidean space … endowed with its standard scalar product. … The book contains 107 figures and the bibliography contains about 89 entries. The book covers an active, interesting and fresh research area. It is very useful for researchers in differential geometry and related subjects." (Kazim Ilarslan, Zentralblatt MATH, Vol. 1149, 2008)