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1 SGC - 70

2 2, 3 23 ɛ-δ [8][14] [1],[2] [4][7] 2 [4]

3 R n

4 iii

5 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

6 2 2.1 R n R n R n 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 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 A A = A

7 3 3.1 V n 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 V n e 1,e 2,...,e n V e i V (i =1, 2,...,n) v = n i=1 vi e i V

8 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 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)

9 5 5.1 X A c := X \ A A X X A, B (A c ) c = A, (A B) c = A c B c, (A B) c = A c B c 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 {A λ } λ Λ ( A λ)c = ( A c λ, A λ)c = A c λ. λ Λ λ Λ λ Λ λ Λ X x Y f(x) 1 (map) f : X Y f X f (domain) Y f (target) R f : X R (function)

10 [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.

11 , 78 27, 77, , 102, , , , 65 27, 78 27, 78 19, , , , , , , , , , 32, 82

12 134 5, 35 6, 46 39, , 39 25, 36, , , , n , , 7, 46 19, , , , , , ,

13 , , , 78 27, 78 2, 125, , , , , , , , , 109,

14 , 33, , , 32, [, ] 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 , 102, 104 k (M) () 107 p (V )() 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

15 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

16 PD 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 ISBN TEL.(03) FAX.(03) sk@saiensu.co.jp C TEL.(03) ()

i Version 1.1, (2012/02/22 24),.,..,.,,. R-space,, ( R- space),, Kahler (Kähler C-space)., R-space,., R-space, Hermite,.

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