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Akivis M., Goldberg V. — Differential Geometry of Varieties with Degenerate Gauss Maps
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Название: Differential Geometry of Varieties with Degenerate Gauss MapsАвторы: Akivis M., Goldberg V. Аннотация: In this book the authors study the differential geometry of varieties with degenerate Gauss maps. They use the main methods of differential geometry, namely, the methods of moving frames and exterior differential forms as well as tensor methods. By means of these methods, the authors discover the structure of varieties with degenerate Gauss maps, determine the singular points and singular varieties, find focal images and construct a classification of the varieties with degenerate Gauss maps.The authors introduce the above mentioned methods and apply them to a series of concrete problems arising in the theory of varieties with degenerate Gauss maps. What makes this book unique is the authors' use of a systematic application of methods of projective differential geometry along with methods of the classical algebraic geometry for studying varieties with degenerate Gauss maps. This book is intended for researchers and graduate students interested in projective differential geometry and algebraic geometry and their applications. It can be used as a text for advanced undergraduate and graduate students. Both authors have published over 100 papers each. Each has written a number of books, including Conformal Differential Geometry and Its Generalizations (Wiley 1996), Projective Differential Geometry of Submanifolds (North-Holland 1993), and Introductory Linear Algebra (Prentice-Hall 1972), which were written by them jointly.
Язык: Рубрика: Математика /Статус предметного указателя: Готов указатель с номерами страниц ed2k: ed2k stats Год издания: 2004Количество страниц: 255Добавлена в каталог: 10.06.2008Операции: Положить на полку |
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137 151 214 24 0-pair 194 2-plane at infinity 120 Abe 172 221 Absolute 126—128 177 181 183 Absolute invariant 5 Absolute invariant of curve 38 Absolute of 127 Absolute of 126 Adjacent points 209 Admissible transformation(s) 18 52 54 Affine analogue of Hartman — Nirenberg cylinder theorem 150 172 Affine connection 18 19 47 Affine frame 25 26 Affine parameter 76 Affine space 25 118 122 133 150 154 164 196 199 203 218 Affine space, structure equations of 26 203 Affine transformation(s) 18 25 Affinely complete hypersurface 154 Akivis xiv—xviii xx 15 46 86—89 101 134 149 154 164 169 176 178 179 184 189 194 199 202 203 219 221—223 Algebra of Cayley's octonions 90 Algebra of complex numbers 90 207 219 Algebra of double numbers 207 219 Algebra of dual numbers 207 219 Algebra of quaternions 90 Algebra of split numbers 207 Algebraic cone 198 Algebraic equation 69 Algebraic fourth-degree surface 68 Algebraic geometry 87 88 Algebraic hypercone 101 Algebraic hypersurface 100 101 197 Algebraic hypersurface with degenerate Gauss map 90 Algebraic variety 42 Allendoerfer xiv 87 223 Almost everywhere differentiable function 50 Almost everywhere differentiable mapping 49 Almost everywhere differentiable variety 49 51 Alternation 42 Analytic function 7 Analytic manifold 5 Anticommutativity 9 Antiinvolutive automorphism 90 Arnold xv 103 223 Asymptotic cone(s) of Grassmannian 60 Asymptotic cone(s) of hypersurface 61 Asymptotic cone(s) of variety 58 Asymptotic curve 58 164 Asymptotic direction 86 154 156 Asymptotic tangent 156 Autopolar simplex 27 Axial point 61 62 Band 164 173 Base form(s) 52 Basic equations of variety 52 Basic equations of variety with degenerate Gauss map 94 151 Basis 1 Basis form(s) of 214 Basis form(s) of curve 28 Basis form(s) of focal line 158 Basis form(s) of Gauss image 93 Basis form(s) of Grassmannian 42 53 63 Basis form(s) of hypersurface 182 Basis form(s) of manifold 16 17 Basis form(s) of Monge — Ampere foliation 92 Basis form(s) of parametric manifold 185 Basis form(s) of torse 138 115 Basis form(s) of variety 52 Basis form(s) of variety with degenerate Gauss map 92 Basis hyperplane 145 Basis natural 15 Basis natural of cotangent space 22 Basis natural of dual space 6 Basis natural of fibration 98 Basis natural of projectivization 24 Basis natural of tangent space 22 Basis natural of vector space 1 20 Basis points of plane 209 Basis points of second normal subspace 57 Basis points of subspace 41 Basis vectors 26 Bejancu 218 227 Beltrametti 89 223 Bianchi xiv 87 223 Bijective mapping 8 Bilinear form 10 Bisecant 44 105 Bisecant variety 44 46 Blaschke xix 58 164 223 224 Block diagonal form 165 Boecher 137 224 Borisenko 88 134 172 224 Bourbaki 208 219 224 Brauner xiv 87 224 Bruce xiv 224 Bryant xx 15 46 216 225 Bundle cotangent 6 Bundle of first-order frames 52 Bundle of hyperplanes 101 136 139—142 Bundle of second fundamental forms 55 Canal hypersurface 186 189 190 Cartan xiv xx 11 46 47 86 87 89 206 225 Cartan lemma 10 Cartan number 15 131 132 156 158 162 217 Cartan test 13 15 130—132 157 158 162 216 217 Cartesian coordinates 116 119 123 Cauchy horizon 176 Caustic 103 186 191 Cayley 66—69 225 Cayley's parameterization 68 Center(s) of bundle of hyperplanes 136 140—142 Center(s) of pencil of cubics 40 Center(s) of pencil of straight lines 164 Center(s) projectivization 24 52 53 Chakmazyan 203 218 222 225 226 Chandrasekhar 177 218 226 Character(s) 14 15 156 Characteristic equation of matrix with respect to matrix 186 Characteristic subspace 102 137 138 Chern xx 15 46 47 63 88 133 134 216 225 226 Chern — Lashof — Hartman — Nirenberg lemma 133 Class of differentiable manifold 5 Class of differentiable mapping 7 Classification of three-dimensional varieties 104—105 134 164 173 Closed contour 203 Closed linear form 5 Closed p-form 12 Closure 71 Cobasis 6 51 Coframe 18 23 Collinear vectors 19 Compact hypersurface 88 Complete matrix algebra 207 Complete noncylindrical hypersurface 163 Complete parabolic variety 126 127 134 Complete regular variety 118 Completely integrable system 13 16 92 95 205 Completely reducible system of matrices 165 Completely reducible variety 165 168 Complex conjugate 2-planes 112 Complex conjugate hypercones 113 Complex conjugate points 113 115 Complex manifold 6 Complex numbers, field of 1 Complex projective geometry 88 Component 165 ff. Conditions for a point to be fixed 31 Cone(s) 64—66 74 102—105 108 127 135 154 156 164 169 172 Conformal space 176 177 184 186 218 Congruence 103 196 199 Conic(s) 35 45 46 79 80 103 107—109 113 120 123 Conic(s) conjugate net 105 Conic(s) curve 188 Conic(s) singular point 188 190 191 Conisecant plane 46 Conjugate net 105 188 Connected hypersurface 149 Connected variety 150 Connection form(s) 19 Connection form(s) of affine connection 201 Connection form(s) of de Sitter space 180 Connection form(s) of normal connection 202 Constant zero curvature 149 Convex hyperquadric 126 Coordinates of tangent vector 6 Correlation 23 80 81 Correlative transformation 80 Cotangent bundle 6 Cotangent space 6 22 Cotangent space, basis of 22 Covariant differential 201 202 Covector 3 Covector field 9 Cubic hypersurface 77 85 88 119 Cubic symmetroid 46 77 103 144 145 Cubic symmetroid, tangent hyperplane to 80 Curvature of line 217 Curvature, form of affine connection 201 Curvature, form of de Sitter space 180—181 Curvature, form of normal connection 202 Curvature, tensor of affine connection 202 204 206 Curvature, tensor of de Sitter space 181 Curvature, tensor, 19 Curve in a projective plane 28 ff. Curve with constant projective curvature 40 Curve with zero projective curvature 40 Curve, basis form of 28 Cyclic generator 189 Cyclic group 21 Cyclic variable 124 Cylinder 64 127 128 149 150 154 163 Cylinder theorem 149—150 172 Cylindrical variety 88 Darboux 47 226 Darboux, hyperquadric 177 192 194 Darboux, mapping 177 184 De Sitter space 176 ff. 218 Degenerate, focus variety 102 Degenerate, Gauss map 64 Degenerate, hyperquadric 145 Degenerate, Riemannian metric 176 Degenerate, second fundamental form 97 99 Degenerate, symmmetric affinor 187 Delanoe 92 133 226 Derivational formulas 2 Determinant submanifold 44 47 145 195 Developable surface 64 87 133 172 Dieudonne 5 19 46 47 97 226 227 Differentiable coordinates 50 Differentiable covector field 9 Differentiable function 2 5 7 9 Differentiable manifold 5 21 46 49 51 Differentiable mapping 7 Differentiable mapping, class of 7 Differential 1-form 2 Differential equations of 1-form 9 Differential equations of absolute invariant 5 Differential equations of covector 3—4 Differential equations of p-form 9 Differential equations of relative invariant 5 Differential equations of relative tensor 4 Differential equations of subspace 12 Differential equations of tensor 4 Differential equations of tensor field 6 Differential equations of vector 3 Differential of function 9 Differential of Gauss map 185 Differential operator 6—7 Differential prolongation 15 Differentiation, exterior 11 Differentiation, exterior, of product 11 Differentiation, relative to secondary parameters 6—7 Dimension of bundle of second fundamental forms 55 Dimension of bundle of tangent hyperplanes 71 Dimension, differentiable manifold 5 Dimension, dual variety 71 72 96 Dimension, frame bundle 6 Dimension, free module 9 Dimension, Grassmannian 59 Dimension, leaf of Monge — Ampere foliation 72 Dimension, osculating subspace to Grassmannian 60 Dimension, projectivization 24 Dimension, second normal subspace 57 Dimension, second osculating subspace 101 Dimension, Segre cone 44 Dimension, Segre variety 44 75 Dimension, tangent bundle 6 Dimension, tangent subspace to Grassmannian 60 Direct product 44 Director variety of cone 65 Director variety of cylinder 150 Discriminant of polynomial 68 distribution 13 202 205 Distribution focus 117 154 157 164 213 Distribution hyperplane 145 Distribution point of cubic 35—37 Distribution straight line 46 Distribution, invariant 19 do Carmo 47 226 Double conic 46 Dual basis 6 Dual coframe 23 97 Dual curve 67 Dual defect of dually nondegenerate variety 90 Dual defect of Segre variety 71 76 Dual defect of tangentially nondegenerate variety 71 Dual defect of variety with degenerate Gauss map 72 89 Dual element of 6 Dual map 70 72 73 Dual space 22 70 82 101 Dual tangent space 6 72 Dual theorem 23 Dual variety 71 96 101 Dual variety of cone 74 Dual variety of hypersurface 74 Dual variety of smooth curve 73 Dual variety of tangentially nondegenerate variety 71 Dual variety of variety with degenerate Gauss map 71 72 Dual vector space 3 Duality principle 22 70 89 Dually degenerate variety 71 72 89 97 101 Dually nondegenerate variety 72 81 90 99 101 Dubrovin 96 227 Duggal 218 227 Edge of regression 102 127 128 188 Eigenvalue 139 140 142 150—152 170 Ein 89 227 Einstein space 181 Einstein summation convention 1 Eisenbud 47 227 Element of dual tangent space 6 Element of tangent bundle 6 Elliptic congruence 211 213 Elliptic pencil of hyperspheres 178 Elliptic space 126—128 Elliptic transformation(s) 27 Embedding 44 Embedding theorem 169 ff.
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