Generalized Curvatures

1. Front Matter

   Pages i-xi

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2. Motivations

   1. Introduction

      Pages 1-10

   2. Motivation: Curves

      Pages 13-28

   3. Motivation: Surfaces

      Pages 29-44

3. Background: Metrics and Measures

   1. Distance and Projection

      Pages 47-56

   2. Elements of Measure Theory

      Pages 57-68

4. Background: Polyhedra and Convex Subsets

   1. Polyhedra

      Pages 71-76

   2. Convex Subsets

      Pages 77-88

5. Background: Classical Tools in Differential Geometry

   1. Differential Forms and Densities on E^N

      Pages 91-95

   2. Measures on Manifolds

      Pages 97-99

   3. Background on Riemannian Geometry

      Pages 101-107

   4. Riemannian Submanifolds

      Pages 109-119

   5. Currents

      Pages 121-125

6. On Volume

   1. Approximation of the Volume

      Pages 129-137

   2. Approximation of the Length of Curves

      Pages 139-141

   3. Approximation of the Area of Surfaces

      Pages 143-150

7. The Steiner Formula

   1. The Steiner Formula for Convex Subsets

      Pages 153-164

   2. Tubes Formula

      Pages 165-175

   3. Subsets of Positive Reach

      Pages 177-186

8. The Theory of Normal Cycles

   1. Invariant Forms

      Pages 189-192

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.

• Gaussian curvature
• Riemannian geometry
• Riemannian manifold
• computational geometry
• computer graphics
• curvature
• curvature measure
• differential geometry
• discrete geometry
• manifold
• submanifold
• triangulation
• visualization

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)

• Institut Camille Jordan, Université Claude Bernard Lyon 1, 69622, France

  Jean-Marie Morvan

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