SGC - 70
2, 3 23 ɛ-δ 2.12.8 3 2.92.13 4 2 3 1 2.1 2.102.12 [8][14] [1],[2] [4][7] 2 [4] 1 2009 8
1 1 1.1... 1 1.2... 4 1.3 1... 8 1.4 2... 9 1.5... 12 1.6 1... 16 1.7... 18 1.8... 21 1.9... 23 2 27 2.1 R n... 27 2.2... 35 2.3... 37 2.4... 44 2.5... 52 2.6... 56 2.7... 61 2.8... 66 2.9... 73 2.10... 75 2.11... 77 2.12... 82 2.13... 84 3 86 3.1... 86 3.2 1... 90 3.3... 93 3.4... 98 3.5 1...102 3.6...104
3.7...107 3.8...115 3.9...120 4 125 4.1...125 4.2...133 4.3...139 4.4...143 4.5...146 4.6 1...149 4.7...154 4.8...160 4.9...163 4.10...170 4.11...174 4.12 2...177 4.13...180 5 185 5.1...185 5.2...188 5.3...189 5.4...196 5.5...197 199 200 iii
1 1.1 I R I ( a, b ), ( a, b ], [ a, b ), [ a, b ] ( a<b + ) c : I R 2 R 2 (curve) (plane curve) c : I R 2 C I c : I R 2 C I Ĩ C c : Ĩ R2 c I = c c : I R 2 t I c (t) c (tangent vector) (velocity vector) c(t) =(x(t),y(t)) c (t) =(x (t),y (t)) c (t) = lim h 0 c(t + h) c(t) h c (t) c(t) c (t) c(t) c 1.1 c (t) c(t) c (t) c(t) 1 3 R n v 1.1
2 2.1 R n R n R n 5.2.3 1 x R n r>0 B(x, r) :={ y R n x y <r}, B(x, r) :={ y R n x y r } B(x, r) (open ball) B(x, r) (closed ball) x B(x, r) B(x, r) (center)r B(x, r) B(x, r) (radius) A R n 2.1.1 x R n A (interior point) r>0 B(x, r) A A A A (interior) 1 x R n A (boundary point) r>0 B(x, r) A A c := X \ A 2.1 A A A (boundary) A A A A A A A A A A A A = A A c (A c )= A 2.1.2 A A = A
3 3.1 V n 3.1.1 ω : V R V V (dual (vector) space) V 1 (1-form) (covector) V V ω, ω 1,ω 2 V, a R, v V (ω 1 + ω 2 )(v) :=ω 1 (v)+ω 2 (v), (aω)(v) :=aω(v). R n v R n v v (u) := u, v (u R n ) 1, R n V, v V v (u) := u, v (u V ) v V ω V ω V ω(u) = ω,u (u V ) V e 1,e 2,...,e n ω := n i=1 ω(e i)e i u = n i=1 ui e i V ω(u) = n i=1 ui ω(e i )= ω,u v V (v ) = v V v v V V V V v V v V 3.1.2 V n e 1,e 2,...,e n V e i V (i =1, 2,...,n) v = n i=1 vi e i V
4 4.1 4.1.1 M M p T p M, p M p, p (Riemannian metric) (0, 2) M, p (p M) (Riemannian manifold) M, p, p p, R 3 M 1 2 M B. R n, R n R n T p R n (p R n ) R n, R n R n (Euclidean metric), R n R n M n, M 4.1.2 v T p M v := v, v 1/2 v (length) v, w T p M (v, w o) v, w = v w cos θ θ [0,π] v w (angle) g p (v, w) := v, w (v, w T p M, p M)
5 5.1 X A c := X \ A A X 5.1.1 X A, B (A c ) c = A, (A B) c = A c B c, (A B) c = A c B c. 5.1.2 Λ λ Λ X A λ A λ (λ Λ) {A λ } λ Λ X {A λ } λ Λ A λ := { x X λ Λ x A λ }, λ Λ A λ := { x X λ Λ x A λ } λ Λ 5.1.3 {A λ } λ Λ ( A λ)c = ( A c λ, A λ)c = A c λ. λ Λ λ Λ λ Λ λ Λ 5.1.4 X x Y f(x) 1 (map) f : X Y f X f (domain) Y f (target) R f : X R (function)
[1] 1 [2] I, II 2, 3 [3] 6 [4] [5] [6] [7] 5 [8] I.M. J.A. [9] 5 [10] 11 [11] 2 [12] R. L.W. [13] J. [14] S. Helgason, Differential geometry, Lie groups, and symmetric spaces, Graduate Studies in Mathematics 34, American Mathematical Society.
85 78 190 190 24 27 31, 78 27, 77, 189 78 99, 102, 104 5 198 32, 82 117 117 81 13 21 165 24 125 119 170, 172 84 190 68 64 5, 65 27, 78 27, 78 19, 140 176 191 30, 75 168 23, 39 127 1 18, 143 16 143 4 43 152 5 5 156 134 133 195 20, 141 87, 194 98 194 55 98 194 56, 181 170 191 2, 133 3 192, 193 192 24, 32, 82
134 5, 35 6, 46 39, 42 37 126 35, 39 25, 36, 39 35 134 39, 190 176 163 72 74, 187 72 147 147 30, 75 13 n 24 13 187 68 198 67 190 190 162 121 169 194 163 2 6 64 64 190 42 46 73 20, 141 73 7 1, 7, 46 19, 52 189 192 198 192 191 92 153 194 86 86 16, 150 18, 150 170 1, 48 182 112, 122 1 8 1 8 1 16, 149 2 11 2 11 2 178 153 194 132 130 93 40 128 3, 10 160 191 191 192 179 195 196 191 197 42 201
93 95 94 94 97 195 32 33 79 187 127 127 127 129 187 42 192 119 162, 191 27, 78 194 194 194 27, 78 27, 78 2, 125, 188 128 120 129 194 32 75 181 182 65 2 2 98 9 32 196 37 107 107, 113 43 58 34, 60 67 193, 195 190 39 188 61 191 27 24 13 145 145 18, 145 27, 79 119 79 1 189 10 52 181 87 16, 149 16 84 65 63 12 18, 52 170 179 9 9, 109, 112 111 112 73 22 202
22 196 174 174 32, 33, 170 189 125 188 188 128 90 90 86 4 181 181 55 134 137 8, 125 130 125 129 128 40 162 162 122 64 64 140 81 81 31, 32, 79 129 [, ] 55 = 34 33, 192, 125, 194, R n 125, 188 R R ω 92 c ω 107, 113 D 19, 140, 142 c X c 18, 143 ξ vx 147 1, 125 R n 188 95 99, 102, 104 k (M) () 107 p (V )() 98 1 86 1 90 1 112 Ā 27, 79 A 27, 78 area( ) () 22 B(p, r) () 137 B(x, r) 27 B(x, r) 27 A 27, 78 D 121 i, 6, 47, 48, 53 x i C 45 C 56 C 67 C 127 D, D p exp, exp p 163 d(p, q) () 134 d area 22 Δf 181 δ j i, δ ij 87, 194 det( ) () 191, 193 df 58, 196 dim 39, 190 div X 181 dω 117 ds 2 1 8, 126 d vol 130 e i R n 9, 49, 190 exp, exp p 163 f T, f ω 98 203
Γij k 20, 141 g ij, g 1 8, 126 grad f 56, 181 H 13 Hess f 181 h ij 2 11 I 1 8 II 2 11 K, K σ 13, 160 κ 1, κ 2 13 L(c) () 2, 133 M f N 129 M/Γ 68 P c(t) () 145 p 98 R(u, v)w () 156 R 3 23 R 3 24 Ric 162 RP n 72 Scal 162 sgn(σ) () 191 S n (r), S n () 5, 40, 65 supp 112, 122 t A 195 TM 73 T n 42 T p S 7 T p M 46 Tp M 90 tr( ) () 191, 193 Tt s (V )((s, t) ) 93 vf 18, 52 vol( ) () 132 V 86 204
1963 1991 PD 1994 2000 2005 Metric geometry The Geometry of Total Curvature on Complete Open Surfaces K. Shiohama, M. Tanaka, Cambridge University Press, 2003 Metric Measure GeometryEuropean Mathematical Society, 2016 SGC -70 2016 5 25 ISBN 978 4 7819 9912 8 2009 11 25 TEL.(03)5474 8816 FAX.(03)5474 8817 http://www.saiensu.co.jp sk@saiensu.co.jp C TEL.(03)5474 8500 () 151 0051 1 3 25