# Generalized Curvatures

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.

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### Contents

 Introduction 2 12 Different Possible Classiﬁcations 3 Motivation 4 Background Metric and Measures 5 Background Classical Tools on Differential Geometry 6 On Volume 7 The Theory of Normal Cycles 8 Applications to Curves and Surfaces 10
 123 Three Theorems 124 On Volume 128 Approximation of the Volume 129 132 A General Evaluation Theorem for the Volume 131 133 An Approximation Result 133 134 A Convergence Theorem for the Volume 135 1342 Statement of the Theorem 137 Approximation of the Length of Curves 139

 Motivations 12 Motivation Curves 13 212 The General Deﬁnition 14 213 The Length of a C¹Curve 15 214 An Obvious Convergence Result 16 22 The Curvature of a Curve 17 222 The Global or Total Curvature 19 23 The Gauss Map of a Curve 21 24 Curves in E² 22 243 The Signed Curvature of a Smooth Plane Curve 24 244 The Signed Curvature of a Plane Polygon 26 245 Signed Curvature and Topology 27 25 Conclusion 28 Motivation Surfaces 29 313 Warning The Lantern of Schwarz 30 32 The Pointwise Gauss Curvature 33 322 Gauss Curvature and Geodesic Triangles 34 323 The Angular Defect of a Vertex of a Polyhedron 36 324 Warning A Negative Result 37 325 Warning The Pointwise Gauss Curvature of a Closed Surface 39 326 Warning A Negative Result Concerning the Approximation by Quadrics 40 33 The Gauss Map of a Surface 41 332 The Gauss Map of a Polyhedron 42 34 The Global Gauss Curvature 43 35 The Volume 44 Background Metrics and Measures 46 Distance and Projection 47 42 The Projection Map 49 43 The Reach of a Subset 52 44 The Voronoi Diagrams 55 Elements of Measure Theory 57 512 Measures 58 514 Signed Measures 59 515 Borel Measures 60 522 Integral of Measurable Functions 61 53 The Standard Lebesgue Measure on Eᴺ 62 531 Lebesgue Outer Measure on R and Eᴺ 63 532 Lebesgue Measure on R and Eᴺ 64 54 Hausdorff Measures 65 55 Area and Coarea Formula 66 56 Radon Measures 67 Background Polyhedra and Convex Subsets 69 Polyhedra 71 62 Euler Characteristic 74 63 Gauss Curvature of a Polyhedron 75 Convex Subsets 77 712 The Support Function 79 713 The Volume of Convex Bodies 80 72 Differential Properties of the Boundary 81 73 The Volume of the Boundary of a Convex Body 82 74 The Transversal Integral and the Hadwiger Theorem 84 742 Transversal Integral 85 743 The Hadwiger Theorem 86 Background Classical Tools in Differential Geometry 90 Differential Forms and Densities on Eᴺ 91 812 Integration of NDifferential Forms on Eᴺ 93 82 Densities 94 822 Integration of Densities on Eᴺ and the Associated Measure 95 Measures on Manifolds 97 92 Density and Measure on a Manifold 98 93 The Fubini Theorem on a Fiber Bundle 99 Background on Riemannian Geometry 101 102 Properties of the Curvature Tensor 102 103 Connexion Forms and Curvature Forms 103 105 The GaussBonnet Theorem 104 107 The Grassmann Manifolds 105 1072 The Grassmann Manifold GNk 106 1073 The Grassmann Manifolds AGNk and AGᴼNk 107 Riemannian Submanifolds 109 112 The Volume of a Submanifold 112 113 Hypersurfaces in Eᴺ 113 1132 kthMean Curvature of a Hypersurface 114 114 Submanifolds in Eᴺ of Any Codimension 115 1142 kthMean Curvatures in Large Codimension 116 1144 The GaussCodazziRicci Equations 117 115 The Gauss Map of a Submanifold 118 Currents 121 122 Rectiﬁable Currents 122
 142 An Approximation by a Polygonal Line 140 Approximation of the Area of Surfaces 143 152 Triangulations 144 1522 Geometric Invariant Associated to a Triangulation 145 154 A Bound on the Deviation Angle 146 1542 Proof of Theorem 45 147 155 Approximation of the Area of a Smooth Surface by the Area of a Triangulation 150 The Steiner Formula 151 The Steiner Formula for Convex Subsets 153 Segments Discs and Balls 155 163 Convex Bodies in Eᴺ Whose Boundary is a Polyhedron 158 164 Convex Bodies with Smooth Boundary 159 165 Evaluation of the Quermassintegrale by Means of Transversal Integrals 161 166 Continuity of the 𝜱k 162 167 An Additivity Formula 164 Tubes Formula 165 172 The Tubes Formula of Weyl 1939 168 1722 Intrinsic Character of the Mk 170 173 The Euler Characteristic 171 175 Transversal Integrals 172 176 On the Differentiability of the Immersions 174 Subsets of Positive Reach 177 182 The Steiner Formula 180 183 Curvature Measures 182 185 The Problem of Continuity of the 𝜱k 184 186 The Transversal Integrals 186 The Theory of Normal Cycles 187 Invariant Forms 189 192 Invariant Differential Forms on Eᴺ x Sᴺ¹ 190 193 Examples in Low Dimensions 192 The Normal Cycle 193 2011 Normal Cycle of a Smooth Submanifold 194 2013 Normal Cycle of a Polyhedron 195 2014 Normal Cycle of a Subanalytic Set 196 203 A Convergence Theorem 198 2031 Boundness of the Mass of Normal Cycles 199 204 Approximation of Normal Cycles 200 Curvature Measures of Geometric Sets 205 2111 The Case of Smooth Submanifolds 206 2112 The Case of Polyhedra 208 212 Continuity of the Mk 209 213 Curvature Measures of Geometric Sets 210 Second Fundamental Measure 213 222 Second Fundamental Measure Associated to a Geometric Set 214 223 The Case of a Smooth Hypersurface 215 224 The Case of a Polyhedron 216 226 An Example of Application 217 Applications to Curves and Surfaces 220 Curvature Measures in E² 221 2322 The Mass of the Normal Cycle of a Domain in E² 223 233 Plane Curves 224 2332 The Mass of the Normal Cycle of a Curve in E² 225 234 The Length of Plane Curves 226 2342 Polygon Lines 227 2352 Polygon Lines 228 Curvature Measures in E³ 231 2422 The Length of Space Curves 232 2423 The Curvature of Space Curves 233 243 Surfaces and Bounded Domains in E³ 234 2432 The Mass of the Normal Cycle of a Domain in E³ 235 2433 The Curvature Measures of a Domain 236 244 Second Fundamental Measure for Surfaces 238 Approximation of the Curvature of Curves 241 252 Curves in E³ 242 Approximation of the Curvatures of Surfaces 249 262 Approximation by a Triangulation 250 2622 Approximation of the Curvatures 251 2623 Triangulations Closely Inscribed in a Surface 252 On Restricted Delaunay Triangulations 253 2712 The Empty Ball Property 254 2713 Delaunay Triangulation Restricted to a Subset 255 272 Approximation Using a Delaunay Triangulation 256 2723 Convergence of the Normals 257 2724 Convergence of Length and Area 258 Bibliography 261 Index 265 Copyright

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