Jean-Marie Morvan 
Generalized Curvatures [PDF ebook] 

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. Let G be the group of rigid motions of E . We say that a 0 quantity Q(S) associated to S is geometric with respect to G if the corresponding 0 quantity Q[g(S)] associated to g(S) equals Q(S), for all g?G . For instance, the 0 diameter of S and the area of the convex hull of S are quantities geometric with respect to G . But the distance from the origin O to the closest point of S is not, 0 since it is not invariant under translations of S. It is important to point out that the property of being geometric depends on the chosen group. For instance, if G is the 1 N group of projective transformations of E , then the property of S being a circle is geometric for G but not for G , while the property of being a conic or a straight 0 1 line is geometric for both G and G . This point of view may be generalized to any 0 1 subset S of any vector space E endowed with a group G acting on it.

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