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i 1 1 1.1.............................. 1 1.2................................. 7 1.3......................... 13 2 23 2.1......................... 23 2.2............................... 31 3 37 3.1 Riemann.............................. 37 3.2............................... 42 4 54 4.1............................. 54 4.2 Crofton............................. 60 4.3............................ 65 4.4 Poincaré............................. 68 4.5 Steiner Hotelling................... 74 4.6 Blaschke............................. 80 5 Euclid 85 5.1 Euclid............... 85 5.2 Euclid Crofton...................... 93 5.3 Euclid........................ 103 5.4 Poincaré............................. 106 5.5 Steiner Hotelling................... 111 6 118 6.1......................... 118 6.2 Crofton......................... 119

1 1 3 Riemann Riemann 1.1 1.1.1 V V R V V V R f, g V r (f + g)(v) f(v) + g(v), (rf)(v) rf(v) (v V ) f + g rf f + g, rf V V δj i { δj i 1, i j 0, i j V {u 1,..., u n } f i (u j ) δ i j V {f i } V dim V dim V {f i } {u j } v V v(f) f(v) (f V ) v : V R v (V ) (V ) V (V ) V 1.1.2 U, V, W f : U V W f v V U W ; u f(u, v)

2 1 u U V W ; v f(u, v) 1.1.3 V V V R ; (f, v) f(v) 1.1.4 V f : V V R f(v 1, v 2 ) f(v 2, v 1 ) (v 1, v 2 V ) v 0 f(v, v) > 0 f V 1.1.5 R 3 (x 1, x 2, x 3 ) (y 1, y 2, y 3 ) (x 2 y 3 x 3 y 2, x 3 y 1 x 1 y 3, x 1 y 2 x 2 y 1 ). R 3 R 3 R 3 R 3 1.3.14 1.1.6 A n R n R n x x t x f A : R n R n R ; (x, y) t xay f A (x, x) x A f A R n R n A f A

1.1. 3 1.1.7 V 1,..., V k, W f : V 1 V k W f 1.1.8 R n n M n (R) R n n det : n {}}{ R n R n M n (R) R 1.1.9 V 1,..., V k v 1 + + v k k j1 V j k V j j1 (v 1,..., v k ) k j1 V j k V j ; v 1 + + v k (v 1,..., v k ) j1 V j (j N) j N V j V j j N V j V j j N V j j N V j ; j N v j (v j ) V j V j j N j N v j j 0 {}}{ 1.1.10 V V V p V p p V p

4 1 p V R φ, ψ p V r (φ + ψ)(g 1,..., g p ) φ(g 1,..., g p ) + ψ(g 1,..., g p ), (rφ)(g 1,..., g p ) rφ(g 1,..., g p ) (g 1,..., g p V ) φ + ψ rφ φ + ψ, rφ p V p V V R + V + 2 V + V 1 V (V ) V 0 V R V i V p V i0 p V A q V B (A B)(g 1,..., g p+q ) A(g 1,..., g p ) B(g p+1,..., g p+q ) (g 1,..., g p+q V ) A B : p+q {}}{ V V R A B V p + q A B A B V u 1,..., u p (u 1 u p )(f 1,..., f p ) f 1 (u 1 ) f p (u p ) (f 1,..., f p V ) u 1 u p : p {}}{ V V R u 1 u p V p 1.1.11 V p V q V p+q V (A, B) A B V V V ( ) ( ) A p B q A p B q (A p p V, B q q V ) p q p,q

1.1. 5 V p {}}{ V V p V (u 1,..., u p ) u 1 u p 1.1.10 A B A B u 1 u p u i 1.1.12 V n u 1,..., u n V ( ) u i1 u ip (1 i 1,..., i p n) p V p V n p u 1,..., u n f 1,..., f n ( ) n a i 1 i p u i1 u ip 0 (a i 1 i p R) i 1,...,i p1 1 k 1,..., k p n k 1,..., k p (f k 1,..., f kp ) a k 1 k p 0 ( ) ( ) p V p V A V n g g g(u i )f i g 1,..., g p V i1 A(g 1,..., g p ) n A g 1 (u i1 )f i 1,..., A i 1 1 n i 1,...,i p1 n i 1,...,i p1 n i 1,...,i p1 n g p (u ip )f ip i p1 g 1 (u i1 ) g p (u ip )A(f i 1,..., f ip ) A(f i 1,..., f ip )(u i1 u ip )(g 1,..., g p ). A(f i 1,..., f ip )u i1 u ip ( ) p V ( ) p V ( ) p V n p

6 1 1.1.13 1.1.12 p V A A n i 1,...,i p1 A(f i 1,..., f ip )u i1 u ip A A(f i 1,..., f ip ) A 1.1.14 V W φ : p {}}{ V V W Φ(v 1 v p ) φ(v 1,..., v p ) (v i V ) Φ : p V W u 1,..., u n V f 1,..., f n 1.1.12 u i1 u ip (1 i 1,..., i p n) p V Φ(u 1 u p ) φ(u 1,..., u p ) Φ p V v i V n n Φ(v 1 v p ) Φ f i 1 (v 1 )u i1 f ip (v p )u ip i 1 1 n i 1,...,i p1 n i 1,...,i p1 i p1 f i 1 (v 1 ) f ip (v p )Φ(u i1 u ip ) f i 1 (v 1 ) f ip (v p )φ(u i1,..., u ip ) n φ f i 1 (v 1 )u i1,..., i 1 1 φ(v 1,..., v p ) n f ip (v p )u ip Φ Φ p V Φ i p1

1.2. 7 1.1.15 V W F : V W p F : p V p W v 1,..., v p V p F (v 1 v p ) F (v 1 ) F (v p ) p V A p F (A)(f 1,..., f p ) A(f 1 F,..., f p F ) (f 1,..., f p W ) p F (A) p W p F f 1,..., f p W p F (v 1 v p )(f 1,..., f p ) (v 1 v p )(f 1 F,..., f p F ) (f 1 F )(v 1 ) (f p F )(v p ) (F (v 1 ) F (v p ))(f 1,..., f p ) p F (v 1 v p ) F (v 1 ) F (v p ) 1.1.12 p V p F 1.2 1.2.1 V p V A 1 i < j p i (t i,j A)(f 1,..., f p ) A(f 1,..., f j,..., f i,..., f p ) (f 1,..., f p V ) t i,j : p V p V p V {A p V t i,j A A (1 i < j p)} j V R + V + 2 V + V {1,..., p} S p p V A q V B (A B)(g 1,..., g p+q ) 1 sgn(σ)(a B)(g σ(1),..., g σ(p+q) ) p!q! σ S p+q (g 1,..., g p+q V )

8 1 A B p+q V A B p+q V A B A B 1.2.2 v 1,..., v p V v 1 v p p V i j t i,j (v 1 v p ) v 1 v j v j i v p. 1.2.3 V r T T (g 1,..., g r ) σ Sr sgn(σ)t (g σ(1),..., g σ(r) ) (g 1,..., g r V ) T r V T r V 1.2.1 A B p+q V p V q V p+q V (A, B) A B C r V (A B) C A (B C) V V u 1,..., u p u 1 u p σ S p sgn(σ)u σ(1) u σ(p) p u i u 1 u p i1 T r V 1 i < j r i j τ S r (t i,j T )(g 1,..., g r ) T (g 1,..., g j,..., g i,..., g r ) i T (g τ(1),..., g τ(r) ) j σ S r sgn(σ)t (g τσ(1),..., g τσ(r) ) sgn(τ) σ S r sgn(τσ)t (g τσ(1),..., g τσ(r) ) σ S r sgn(σ)t (g σ(1),..., g σ(r) ) T (g 1,..., g r ). t i,j T T T r V

1.2. 9 A p V B q V A B p+q V A B p+q V p V q V p+q V (A, B) A B (A, B) A B ( 1.1.11) T T A p V, B q V, C r V (A B) C A (B C) S p+q {τ S p+q+r τ(i) i (p + q + 1 i p + q + r)} ((A B) C)(g 1,..., g p+q+r ) 1 sgn(σ)(a B)(g σ(1),..., g σ(p+q) ) C(g σ(p+q+1),..., g σ(p+q+r) )) (p + q)!r! σ S p+q+r 1 sgn(σ) (p + q)!r! σ S p+q+r 1 sgn(τ)a(g στ(1),..., g στ(p) ) B(g στ(p+1),..., g στ(p+q) ) p!q! τ S p+q C(g σ(p+q+1),..., g σ(p+q+r) ) 1 1 sgn(στ) p!q!r! (p + q)! τ S p+q σ S p+q+r (A(g στ(1),..., g στ(p) ) B(g στ(p+1),..., g στ(p+q) )) C(g στ(p+q+1),..., g στ(p+q+r) ) 1 sgn(σ) p!q!r! σ S p+q+r (A(g σ(1),..., g σ(p) ) B(g σ(p+1),..., g σ(p+q) )) C(g σ(p+q+1),..., g σ(p+q+r) ). (A (B C))(g 1,..., g p+q+r ) 1 sgn(σ) p!q!r! σ S p+q+r A(g σ(1),..., g σ(p) ) (B(g σ(p+1),..., g σ(p+q) ) C(g σ(p+q+1),..., g σ(p+q+r) ))

10 1 (A B) C A (B C) σ S p sgn(σ 1 ) sgn(σ) V u 1,..., u p (u 1 u p )(g 1,..., g p ) σ S p sgn(σ)u 1 (g σ(1) ) u p (g σ(p) ) σ S p sgn(σ)u σ 1 (1)(g 1 ) u σ 1 (p)(g p ) σ S p sgn(σ 1 )u σ 1 (1)(g 1 ) u σ 1 (p)(g p ) σ S p sgn(σ)u σ(1) (g 1 ) u σ(p) (g p ) σ S p sgn(σ)(u σ(1) u σ(p) )(g 1,..., g p ). u 1 u p σ S p sgn(σ)u σ(1) u σ(p) 1.2.4 V p {}}{ V V p V (u 1,..., u p ) u 1 u p u 1,..., u p V 1 i < j p u 1 i j u j u i u p u 1 u p p A (A i j) v j p A i ju i i1 v 1 v p (det A)u 1 u p

1.2. 11 1.1.11 (u 1,..., u p ) u 1 u p u 1,..., u p V 1 i < j p u 1 i j u j u i u p u 1 u p i j τ S p u 1 u τ(1) u τ(p) i j u j u i u p σ S p sgn(σ)u τσ(1) u τσ(p) sgn(τ) σ S p sgn(τσ)u τσ(1) u τσ(p) σ S p sgn(σ)u σ(1) u σ(p) u 1 u p. u 1 i j u j u i u p u 1 u p u 1,..., u p u 1 u p 0 i j u i u j u 1 u p u i u j 1 u i u j u 1 u p 1 0 σ u σ(1) u σ(p) sgn(σ)u 1 u p S p σ S p p A (A i j)

12 1 v j p A i ju i i1 v 1 v p ( p ) A i 1 1 u i1 i 1 1 p A ip p u ip i p1 σ S p A σ(1) 1 u σ(1) A σ(p) p u σ(p) ( 0 ) sgn(σ)a σ(1) 1 A σ(p) u 1 u p σ S p (det A)u 1 u p. p 1.2.5 u 1,..., u n V ( ) u i1 u ip (1 i 1 < < i p n) ( ) n p V dim( p V ) p u 1,..., u n f 1,..., f n ( ) a i1 ip u i1 u ip 0 (a i1 ip R) i 1 < <i p 1 k 1 < < k p n k 1,..., k p (f k 1,..., f kp ) a k1 k p 0 u i1 u ip ( ) p V p V A g 1,..., g p V A(g 1,..., g p ) n A g 1 (u j1 )f j 1,..., j 1 1 n j 1,...,j p1 n j 1,...,j p1 j 1 < <j p j 1 < <j p n g p (u jp )f jp j p1 g 1 (u j1 ) g p (u jp )A(f j 1,..., f jp ) A(f j 1,..., f jp )(u j1 u jp )(g 1,..., g p ) A(f jσ(1),..., f jσ(p) )(u jσ(1) u jσ(p) )(g 1,..., g p ) σ S p sgn(σ)a(f j1,..., f jp )(u jσ(1) u jσ(p) )(g 1,..., g p ) σ S p j 1 < <j p A(f j1,..., f jp )(u j1 u jp )(g 1,..., g p ).

1.3. 13 A u i1 u ip j 1 < <j p A(f j1,..., f jp )u j1 u jp p V ( ) p V ( ) p V ( ) n p 1.2.6 V W F : V W 1.1.15 p F : p V p W p F ( p V ) p W p F : p V p W 1.2.1 t V i,j : p V p V, t W i,j : p W p W p F p F t V i,j t W i,j p F p F p F ( p V ) p W p F : p V p W 1.3 1.3.1 V V A V V α A(x, y) (α(x))(y) (x, y V ) V 2 V V V Hom(V, V ) α Hom(V, V ) 2 V A 0 x V (α(x))(x) > 0 A V α Hom(V, V ) A(x, y) (α(x))(y) (x, y V ) A 2 V A 2 V α Hom(V, V ) 2 V Hom(V, V ) A 0 x V (α(x))(x) > 0 A(x, x) > 0 A V 1.3.2 V,, 1.3.1 Hom(V, V ) α p V ( p V ) 1.2.6 α : V V p α : p V p V ( p V ) 2 ( p V ) p V

14 1 p V ( p V ) p V φ ( p V ) Φ φ(v 1,..., v p ) Φ(v 1 v p ) (v 1,..., v p V ) p α 2 ( p V ) A V u 1,..., u p v 1,..., v p A(u 1 u p, v 1 v p ) ( p α(u 1 u p ))(v 1 v p ) (α(u 1 ) α(u p ))(v 1 v p ) (α(u 1 ) α(u p ))(v 1,..., v p ) σ S p sgn(σ)(α(u σ(1) ) α(u σ(p) ))(v 1,..., v p ) σ S p sgn(σ)(α(u σ(1) ))(v 1 ) (α(u σ(p) ))(v p ) σ S p sgn(σ) u σ(1), v 1 u σ(p), v p det( u i, v j ) 1 i,j p. u 1,..., u n V 1.2.5 u i1 u ip (1 i 1 < < i p n) p V 1 i 1 < < i p n 1 j 1 < < j p n A(u i1 u ip, u j1 u jp ) δ i1 j 1 δ ipj p A p V 1.3.3, V p V 1.3.2 A A, u u, u 1.3.4 1.3.2 V u 1,..., u p v 1,..., v p u 1 u p, v 1 v p det[ u i, v j ] 1 i,j p V e 1,..., e n p V e i1 e ip (1 i 1 < < i p n)

1.3. 15 1.3.5 1.3.4 V u 1, u 2 u 1 u 2 2 u 1, u 1 u 1, u 2 u 1 u 2, u 1 u 2 u 2, u 1 u 2, u 2 u 1 2 u 2 2 u 1, u 2 2 u 1 u 2 θ u 1 u 2 u 1 2 u 2 2 sin 2 θ u 1 2 u 2 2 (1 cos 2 θ) u 1 2 u 2 2 u 1, u 2 2 u 1 u 2 2 u 1 u 2 u 1 u 2 2 V V 1.3.6 R n u u m n R n u 1,..., u m, v 1,..., v m u i [u ij ], v i [v ij ] u 11 u 1n.. v 11 v m1.. u 1. [v 1 v m] [u i v j ] [ u i, v j ]. u m1 u mn v 1n v mn u m m 1.3.4 u 11 u 1n v 11 v m1 det.... det[ u i, v j ] u m1 u mn v 1n v mn u 1 u m, v 1 v m. R n e 1,..., e n u i [u i1... u in ] n u ij e j. j1 ( 1.2.4) ( n ) ( n ) u 1 u m u 1j1 e j1 u mjm e jm j 1 1 #{j 1,...,j m}m j 1 < <j m j m1 u 1j1 u mjm e j1 e jm u 1j1 u 1jm.. u mj1 u mjm e j1 e jm

16 1 v 1 v m j 1 < <j m v 1j1 v 1jm.. v mj1 v mjm e j1 e jm u 1 u m, v 1 v m j 1 < <j m u 1j1 u 1jm.. u mj1 u mjm v 1j1 v 1jm.. v mj1 v mjm det u 11 u 1n.. v 11 v m1.. j 1 < <j m j 1 < <j m u m1 u mn u 1j1 u 1jm.. u mj1 u mjm u 1j1 u 1jm.. u mj1 u mjm v 1n v mn v 1j1 v 1jm.. v mj1 v mjm v 1j1 v mj1.. v 1jm v mjm m n 1.3.7 V, V u 1,..., u p p u 1 u p u i u 1,..., u p i1 u i v i w i v 1 0, w 1 u 1 v i 1, w i 1 v i w i u i v i + w i, v i span{u 1,..., u i 1 }, w i span{u 1,..., u i 1 } v i w i w i 2 u i 2 u 1 u i w 1 w i (1 i p)

1.3. 17 1.3.4 u 1 u p w 1 w p u 1 u p 2 w 1 w p 2 w 1 w p, w 1 w p det( w i, w j ) w 1, w 1 0 0 det 0.............. 0 0 0 w p, w p p p p w i, w i u i, u i u i 2. i1 u 1 u p i1 p u i i w i, w i u i, u i v i 0 u 1,..., u p 1.3.8 V W m n (m n) F : V W JF sup{ F (u 1 ) F (u n ) u 1,..., u n V } F JF 0 F (ker F ) v 1,..., v n i1 i1 JF n F (v 1 v n ) v 1 v n F (v 1) F (v n ) v 1 v n F dim(imf ) < n V u 1,..., u n F (u 1 ),..., F (u n ) F (u 1 ) F (u n ) 0 JF 0

18 1 F (ker F ) v 1,..., v n (ker F ) u 1,..., u n (a ij ) n v j a ij u i 1.2.4 i1 v 1 v n det(a ij )u 1 u n, n F (v 1 v n ) det(a ij ) n F (u 1 u n ). n F (v 1 v n ) v 1 v n det(a ij) n F (u 1 u n ) det(a ij ) u 1 u n n F (u 1 u n ) u 1 u n n F (u 1 u n ). 1.3.4 u 1 u n 1 JF n F (u 1 u n ) n F (v 1 v n ). v 1 v n V w 1..., w n w i w 1 i + w 2 i, w 1 i (ker F ), w 2 i ker F w 1 i w i 1 n F (w 1 w n ) F (w 1 ) F (w n ) F (w 1 1) F (w 1 n) n F (w 1 1 w 1 n). w 1 1,..., w 1 n 1.2.4 n F (w 1 1 w 1 n) 0 (ker F ) n F (w 1 1 w 1 n) w 1 1 w 1 n n F (u 1 u n ). 1.3.7 n F (w 1 w n ) n F (w1 1 wn) 1 w1 1 wn 1 n F (u 1 u n ) w1 1 wn 1 n F (u 1 u n ) n F (u 1 u n ).

1.3. 19 JF n F (u 1 u n ) n F (v 1 v n ). v 1 v n JF n F (v 1 v n ) v 1 v n JF n F (v 1 v n ) v 1 v n JF. 1.3.9 V V 0, V 1 v 0, v 1 V 0 V 1 F JF v 0, v 1 x V 0 x (x x, v 1 v 1 ) + x, v 1 v 1, (x x, v 1 v 1 V 1, x, v 1 v 1 V 1 ) F (x) x x, v 1 v 1 (x V 0 ) V 0 V 1 dim(v 0 V 1 ) dim V 0 1 dim V 1 1 V 0 V 1 F (x) x v 1 ±v 0 JF 1 v 0, v 1. dim(v 0 V 1 ) dim V 0 1 dim V 1 1 V 0 V 1 u 1,..., u p 1 u 1,..., u p V 0 { u i (1 i p 1), F (u i ) u p u p, v 1 v 1 (i p) [ F (u i ), F (u j ) ] 1 0 0 0...................... 1 0 0 0 1 u p, v 1 2.

20 1 JF 2 F (u 1 ) F (u p ) 2 det[ F (u i ), F (u j ) ] 1 u p, v 1 2. (V 0 V 1 ) 2u p, v 0 (V 0 V 1 ) v 1 (V 0 V 1 ) u p, v 0 v 1 u p, v 1 u p + v 0, v 1 v 0. 1 u p, v 1 2 + v 0, v 1 2 JF 2 v 0, v 1 2 1.3.10 1.3.8 m n F 0 : V 0 V V 0 V J(F F 0 ) JF JF 0 F F F 0 1.3.8 J(F F 0 ) 0 JF JF 0 F 1.3.8 V 0 v 1,..., v n J(F F 0 ) F F 0(v 1 ) F F 0 (v n ) v 1 v n F F 0(v 1 ) F F 0 (v n ) F 0(v 1 ) F 0 (v n ) F 0 (v 1 ) F 0 (v n ) v 1 v n JF JF 0. 1.3.11 V, n V e 1,..., e n e 1 e n n V 1.2.4 e 1,..., e n e 1 e n V 1.2.5 n V Re 1 e n 0 k n ξ k V η n k V ξ η n V ξ η B(ξ, η) R ξ η B(ξ, η)e 1 e n 1.2.5 dim k V B : k V n k V R ( ) ( ) n n dim n k V k n k {1,..., n} k I {i 1,..., i k } (i 1 < i 2 < < i k )

1.3. 21 e I e i1 e ik k V {1,..., n} I I c e I c n k V 1.2.4 e I e I c ±e 1 e n B(e I, e I c) ±1 (I {1,..., n}, #I k) B : k V n k V R ξ, ξ B(ξ, ξ ) (ξ, ξ k V ) : k V n k V ξ, ξ k V ξ, ξ e 1 e n B(ξ, ξ )e 1 e n ξ ξ 1.3.12 V, n e 1,..., e n V 1 i 1 < < i k n {i 1,..., i k } c {j 1,..., j n k } 1 j 1 < < j n k n ( ) 1............ n (e i1 e ik ) sgn e j1 e jn k i 1... i k j 1... j n k : k V n k V ξ, η k V (1) ξ η ξ, η e 1 e n, (2) ( ξ) ( 1) k(n k) ξ. 1.3.11 (1) 1.2.4 ( ) 1............ n e i1 e ik e j1 e jn k sgn e 1 e n i 1... i k j 1... j n k 1 l 1 < < l n k n e i1 e ik e l1 e ln k 0 1.3.4 (1) ( ) 1............ n (e i1 e ik ) sgn e j1 e jn k i 1... i k j 1... j n k ( (e i1 e ik )) ( ) 1............ n sgn (e j1 e jn k ) i 1... i k j 1... j ( n k ) ( ) 1............ n 1............ n sgn sgn e i1 e ik. i 1... i k j 1... j n k j 1... j n k i 1... i k

22 1 ( ) ( ) 1............ n 1............ n sgn sgn i 1... i k j 1... j n k j 1... j n k i 1... i ( ) ( k ) i1... i k j 1... j n k 1............ n sgn sgn 1............ n j 1... j n k i 1... i ( ) k i1... i k j 1... j n k sgn ( 1) k(n k) j 1... j n k i 1... i k ( (e i1 e ik )) ( 1) k(n k) e i1 e ik ξ k V ( ξ) ( 1) k(n k) ξ 1.3.13 V, 2 e 1, e 2 V ( ) ( ) 1 2 1 2 e 1 sgn e 2 e 2, e 2 sgn e 1 e 1 1 2 2 1 : V V π/2 1.3.14 V, 3 e 1, e 2, e 3 V ( ) 1 2 3 (e 1 e 2 ) sgn e 3 e 3, 1 2 3 ( ) 1 2 3 (e 2 e 3 ) sgn e 1 e 1, 2 3 1 ( ) 1 2 3 (e 3 e 1 ) sgn e 2 e 2 3 1 2 V V V ; (u, v) (u v) 3

23 2 2.1 2.1.1 X U {U α } α A X X x x V {α A V U α } U 2.1.2 X U {U α } α A X U α X α A U X U {U α } α A V {V i } i I X i I α A V i U α V U 2.1.3 X U {U α } α A X X K {α A K U α } {U α } α A X x K x V x {α A V x U α }

24 2 {V x } x K K K x 1,..., x k K {V xi 1 i k} K { } k k {α A V xi U α } α A V xi U α i1 i1 K k V xi i1 {α A K U α } 2.1.4 Hausdorff X X 2.1.5 Hausdorff X Hausdorff U {U α } α A X X A {α 1,..., α N } A U {U αi 1 i N} X U U U U U X 2.1.6 X (1) X X (2) B {B α } α A X X O x x B α O B α B X X X (3) X X σ (4) X X Lindelöf

2.1. 25 2.1.7 R n R n Lindelöf 2.1.8 X {U α } α A X α A V α U α X {V α } α A X p O p Ōp {U α } α A X α A p U α O p Ōp U α {O p } p X X p X α A Ōp U α Ōp X {O p } p X {W i } i I α A I α I α {i I Wi U α } V α i I α W i V α X {V α } α A X {W i } i I X p 0 X i 0 I p 0 W i0 {W i } i I {O p } p X W i0 q X W i0 O q {O p } p M α 0 A Ōq U α0 p 0 W i0 W i V α0 i I α0 {V α } α A X α A V α U α p V α {W i } i I p O ( ) {i I O W i } ( ) {i 1,..., i k } p V α N O p N N i I α W i. {i 1,..., i k } i I α N W i O W i

26 2 k N j1 W ij k k p W ij j1 j1 W ij U α V α U α 2.1.9 X K K V V x K x C x C x U x U x x U x C x U x {U x x K} K K x 1,..., x N K {U xi 1 i N} K N V V K i1 U xi N N V U xi i1 V 2.1.10 Hausdorff σ X Hausdorff σ X X {K i } i N X {K i } i N N X 2.1.5 X N N X V 1 V 2 V 3 2.1.9 K 1 V 1 V 1 K 2 V 1 K 2 V 1 V 2 V 2 K j+1 V j V j+1 V j+1 V j j i j K i V j {V j } j N X j N V j V j+1 i1 C xi

2.1. 27 {V j } j N X {O j } j N {F j } j N O 1 V 2, O 2 V 3 j 3 O j V j+1 V j 2 F 1 V 1 j 2 F j V j V j 1 O j F j F 1 V 1 V 2 O 1, F 2 V 2 V 1 V 2 V 3 O 2, F j V j V j 1 V j+1 V j 2 O j (j 3) j N F j O j j F i V j F i V i X {O j } j N X i N i N O i i j 3 O i O j X X U {U α } α A i N x F i x U α F i O i x U x,i U x,i U α O i {U x,i x F i } F i F i x 1,i,..., x ai,i F i {U xk,i,i 1 k a i } F i i N U xk,i,i B {U xk,i,i i N, 1 k a i } B X U xk,i,i α U xk,i,i U α B U i N U xk,i,i O i i j 3 O i O j U xk,i,i U xl,j,j i1 {B B B U xk,i,i } {U xl,j,j i j 2} U xk,i,i B B X X 2.1.11 X Y X X O X Y O Y {O Y O O X } X O X B X B Y {B Y B B X }

28 2 O Y O Y O O X O Y x O Y x O x B O B B X x B Y O Y B Y O Y O X O Y Y 2.1.12 R n Q Q n {(q 1,..., q n ) q i Q} R n Q n Q n Q n x (x 1,..., x n ) R n r > 0 B(x; r) {y R n y x < r} B(x; r) B(x; r) R n B {B(q; 1/m) q Q n, m N} R n B B B R n O R n x O O r > 0 B(x; r) O 1/m < r/2 m N Q n R n q Q n q B(x; 1/m) x B(q; 1/m) B(x; 2/m) B(x; r) O B R n R n 2.1.13 R n 2.1.12 R n 2.1.11 2.1.14 Lindelöf X X B {B i } i N N X {U α } α A x U α

2.1. 29 x X, α A x B i U α B i i N i(x, α) N {i(x, α) x X, α A, x U α } N i(x, α) {B i } i N X i N B i U α α A α A α(i) B i U α(i) {U α(i) } i N X X Lindelöf 2.1.15 Lindelöf σ X Lindelöf x X x C x C x U x U x x U x C x U x {U x } x X X X Lindelöf N X {U x } x N X X {U x } x N X σ 2.1.16 X (1) X (2) X Lindelöf (3) X σ (4) X X R n (1) (2) 2.1.14 (2) (3) 2.1.15 Lindelöf σ (3) (1) X σ {K i } i N X {K i } i N K i R n K i {U j } U j R n {U j } X 2.1.13 U j U j X X (1) (2) (3) (1) (3) (4)

30 2 (1), (3) (4) 2.1.10 σ {B i } i I X {O j } j J X O j X x O j x B i(j) O j i(j) I j, k J j k O j O k B i(j) B i(k) i(j) i(k) i : J I J I X (4) (3) X X x X x C x C x U x {U x } x X X X {U x } x X {V α } α A α A x X V α U x V α C x V α {V α } α A X X α A V α V α A X σ A α 0 A A 1 {α A {α s } t s0 A V αs V αs+1, α t α} A 2 A A 1 X 1 α A 1 V α, X 2 α A 2 V α X 1 X 2 X A A 1 A 2 X X 1 X 2 X 1 X 2 X 1 X 2 X 1 X 2 β 1 A 1, β 2 A 2 V β1 V β2 β 1 A 1 V β1 V β2 β 2 A 1 β 2 X 1 X 2 V α0 X 1 X 1 X X 2 X X 1 A A 1 A {α A {α s } t s0 A V αs V αs+1, α t α} α A n(α) min{t 0 {α s } t s0 A V αs V αs+1, α t α}

2.2. 31 A t 0 {α A n(α) t} t0 {α A n(α) t} t {α A n(α) 0} {α 0 } {α A n(α) t} {α A n(α) t + 1} α A A α {β A V β V α } V α 2.1.3 {β A V β V α } A α {α A n(α) t} {α A n(α) t} {β 1,..., β k } {α A n(α) t + 1} A βj {α A n(α) t + 1} t 0 {α A n(α) t} A {α A n(α) t} t0 k j1 A βj 2.2 C C

32 2 2.2.1 X f suppf {x X f(x) 0} 2.2.2 M {U α } α A M C {f α } α A (1) (3) {f α } α A {U α } α A (1) α A 0 f α 1, (2) α A suppf α U α, (3) p M α A f α (p) 1. 2.2.3 2.2.2 (3) {U α } α A p M V {α A V U α } {α 1,..., α k } α A suppf α U α V 0 f α p p V f α (x) α A k f αi (x) (x V ) i1 2.2.4 M {U α } α A M α A Ūα {U α } α A {f α } α A 2.2.5 2.2.4 {U α } α A M {O i } i I i I Ōi M {O i } i I {U α } α A α A i I U α O i Ūα Ōi Ōi Ūα 2.2.4

2.2. 33 2.2.6 (1) R(t) t lim t +0 R(t)e 1/t 0 (2) a(t) a(t) { 0 (t 0) e 1/t (t > 0) a(t) C (3) 0 < u < v t v v t b(t) 0, v < t < u u < t < v 0 < b(t) < 1, u t u b(t) 1 C b(t) (1) t > 0 s 1/t R(t)e 1/t R(t) e 1/t R(1/s) e s R(1/s) s P (s)/q(s) R(1/s) (P (s), Q(s) s ) P (s), Q(s) m, n e s k0 k 0 < s e s > s k /k! N > max{m, n} 0 R(1/s) e s P (s) P (s) Q(s)e s (N n)! Q(s) s (N P (s)/s m N n n)!sm s N Q(s)/s n. P (s)/s m Q(s)/s n s k k! s m P (s)/s m lim s + s N Q(s)/s n 0 lim R(1/s) e s 0 s + lim t +0 R(t)e 1/t 0

34 2 (2) 0 n t R n (t) ( n) d n a dt n (t) { 0 (t 0) R n (t)e 1/t (t > 0) n a C R 0 (t) 1 ( 0) a ( n) ( n + 1) d n a t 0 0 ( n) dtn lim t 0 lim t +0 1 d n a t dt (t) 0 n 1 d n a (t) lim t dtn t +0 1 t R n(t)e 1/t 0. ((1) ) dn a 0 0 ( n) dtn d n+1 a dt (t) d d n a (t) 0 (t < 0), n+1 dt dtn d n+1 a dt (t) d d n a n+1 dt dt (t) n ( d dt R n(t) + R n (t) 1 t 2 R n+1 (t) d dt R n(t) + R n (t) 1 t 2 ) e 1/t (t > 0) R n+1 (t) t ( n+1) (3) (2) a(t) a(t)a(1 t) 0 (t 0), > 0 (0 < t < 1), 0 (1 t) t / 1 a 1 (t) a(s)a(1 s)ds a(s)a(1 s)ds 0 0 ( ) t + v 0 0 0 1 1 1 a 1 ( v u) v t v 0 v u u 1 a 1 v u

2.2. 35 u 1 u v v 0 ( ) ( ) t + v v t b(t) a 1 a 1 v u v u b(t) 2.2.7 M K K U M C f M 0 f 1 K f > 0 U f 0 p K V p V p V p U V p (x 1,..., x n ) x i (p) 0 (1 i n) [ v, v] x i (V p ) (1 i n) v > 0 W p {x V p x i (x) < v (1 i n)} p W p W p V p U 0 < u < v u u, v 2.2.6 (3) C b(t) { b(x 1 (q))b(x 2 (q)) b(x n (q)) (q V p ), f p (q) 0 (q / V p ) f p M C {q M f p (q) 0} W p 0 b(x i (q)) 1 0 f p 1 {W p } p K K K K p 1,..., p k K W p1 W p2 W pk f 1 k (f p 1 + f p2 + + f pk ) f 0 f 1 M C {q M f(q) 0} W p1 W p2 W pk U. K f > 0 U f 0

36 2 2.2.4 2.1.8 α A W α U α M {W α } α A V α W α M {V α } α A Ūα V α 2.2.7 M C g α M 0 g α 1 V α g α > 0 W α g α 0 suppg α W α U α. {U α } α A p M p U {α A U U α } α A g α U 0 g(p) α A g α (p) (p M) g g M C α A g α 0 V α g α > 0 {V α } α A M M g > 0 f α g α /g {f α } α A {U α } α A

37 3 3.1 Riemann Riesz ( 3.1.7) 3.1.13 Riemann Riemann 3.1.1 X 2 X [0, ] µ µ X (1) µ( ) 0 (2) X {A i } A i1 A i A 2 X µ(a) µ(a i ). i1 3.1.2 µ X X A µ(t ) µ(t A) + µ(t A) T 2 X A X µ 3.1.3 f µ X S [, ] µ(x S) 0 [, ] O f 1 (O) X µ f µ 3.1.4 R n Lebesgue Lebesgue Fubini 3.1.5 X σ Borel µ X X Borel µ A X Borel B A B µ(a) µ(b) µ Borel

38 3 3.1.6 X Hausdorff X Borel µ K X µ(k) < µ Radon 3.1.7 (Riesz ) X Hausdorff X K(X) K(X) L : K(X) R (1) f 0 f K(X) L(f) 0 (2) K X sup{l(f) f K(X), f 1, suppf K} < Radon µ X L(f) fdµ (f K(X)) X 3.1.8 C 2.1.16 3.1.7 Riesz Riemann Riemann 3.1.9 M x M M x T x M g x M (U; x 1,..., x n ) ( ) U x g x, x i x j 1 i, j n C g (g x ) x M M Riemann Riemann (M, g) Riemann 3.1.10 Euclid R n x M T x R n R n T x M Riemann R n Riemann Riemann

3.1. Riemann 39 3.1.11 Riemann M M p V p V p V p {V p } p M M M {V p } p M M {U α } α A V p U α V p Ūα 2.2.4 {U α } α A {f α } α A U α x 1,..., x n (g α ) x ( x i, ) δ i,j (x U α ) x j x U α T x M (g α ) x { f α (x)(g α ) x (x U α ), (h α ) x 0 (x / U α ) h α ((h α ) x ) x M α A h α A h α x M x α A x (U; x 1,..., x n ) ( ) U x h x, x i x j 1 i, j n C h M Riemann 3.1.12 ( ) U R n R n U x 1,..., x n y 1,..., y n U f ( ) f(x)dx 1 dx n f(y) det xi dy 1 dy n. y j U U 3.1.13 (M, g) Riemann M (U; x 1,..., x n ) x U n T x (M) Riemann suppf U f K(M) L(f) f(x 1,..., x n ) x 1 x n dx 1 dx n U

40 3 L(f) Euclid Lebesgue Riemann L(f) K(M) L : K(M) R L 3.1.7 Radon µ M L(f) fdµ (f K(M)) M µ M Riemann Riemann Riemann µ µ (M,g) Riemann µ M vol(m) µ M (M) vol(m) M M 1 L(M) M 2 A(M) M 3 V (M) suppf U f K(M) L(f) U y 1,..., y n y l 1.2.4 n k1 det y 1 y n x k y l ( xi y j x k ) x 1 x n f(x) x U 1 x n dx 1 dx n f(y) ( ) x U 1 x n det xi dy 1 dy n y j ( ) f(y) det xi y U j x 1 x n dy 1 dy n f(y) y 1 y n dy 1 dy n U L(f)

3.1. Riemann 41 K(M) L x M Ūx x U x {U x } x M M M {U x } x M {V α } α A Ūx V α 2.2.4 {V α } α A {f α } α A f K(M) f V α f α f V α 0 α f α f 0 ( ) f 1 f f α f f α f α A α f α f V α U x L(f α f) L(f) α A L(f α f) L(f) L {W β } β B M W β M {g β } β B {W β } β B α A f α f β B g β f α f L Euclid L(f α f) β B L(g β f α f) α A L(f α f) α A L(g β f α f). β B L(f α g β f). β B L(g β f) β B α A α A L(f α f) β B L(g β f) L L Riesz Euclid

42 3 3.1.14 n n Riemann 3.1.15 M Riemann (U; x 1,..., x n ) M U µ M φ µ M φ 0 φdµ M φ(x 1,..., x n ) x 1 x n dx 1 dx n U U 3.2 Fenchel 3.2.1 f : M N M N C x M df x : T x (M) T f(x) (N) x f M y N f(x) y f x y f N 3.2.2 (Sard ) U R n f : U R p C f C R p Lebesgue µ µ(f(c)) 0 Sard 3.2.3 f : M N Riemann M Riemann N C f C µ N (f(c)) 0 M N M N {U i } {V i } i (1) U i M (2) V i N (3) f(u i ) V i

3.2. 43 C i C U i V i N (V i ; x 1,..., x p ) V i R p R p Lebesgue µ 3.2.2 µ(f(c i )) 0 x 1 x p V i V i ( V i ) A sup V i 3.1.15 µ N (f(c i )) 0 x 1 x p 0 µ N (f(c i )) Aµ(f(C i )) 0 0 µ N (f(c)) i µ N (f(c i )) 0, µ N (f(c)) 0 3.2.4 m n f : M N m Riemann M n Riemann N C x M 1.3.8 J Jf(x) Jdf x 3.2.5 ( ) m n f : M N m Riemann M n Riemann N C φ M µ M N y φ(x)dµ f 1 (y)(x) N f 1 (y) µ N φjf M µ M φ 0 ( ) φ(x)dµ f 1 (y)(x) dµ N (y) φ(x)jf(x)dµ M (x) f 1 (y) M N x M Vx n (M) T x (M) n Stiefel V n (M) Vx n (M) x M V n (M) V n (M) M Stiefel V n (M) V n (M) R ; (u 1,..., u n ) df(u 1 ) df(u n )

44 3 Jf(x) sup{ df(u 1 ) df(u n ) (u 1,..., u n ) V n x (M)} M R O {x M Jf(x) 0} M O µ M φ(x)jf(x)dµ M (x) φ(x)jf(x)dµ M (x) M x O x U x f(u x ) f(x) f : U x f(u x ) Euclid f Euclid N R n F R m n M N F f : M N F N; (y, t) y y 1,..., y n N R n x 1,..., x m M N F R m y i f x i (1 i n) x n+1,..., x m F φ M φ(y, t) x n+1 x m M y 1 y n f N M Fubini y φ(y, t) x F n+1 x m M y 1 y n fdx n+1 dx m (t) N ( φ(y, t) ) x F n+1 x m dx n+1 dx m (t) M y 1 y n ( ) φ(x)dµ f 1 (y)(x) f 1 (y) y 1 y n N φ(y, t) x M n+1 x m M y 1 y n fdx 1 dx m N ( ) φ(x)dµ f 1 (y)(x) N f 1 (y) y 1 y n dy 1 dy n (y) N ( ) φ(x)dµ f 1 (y)(x) dµ N (y) f 1 (y) N O N N

3.2. 45 1 i n M N F F x i ( ) ( ) + x i x i x i ( ) (( ) df df x i x i F F F. ). y 1 y n ( ) N ( ) N df df x 1 x n (( ) ) (( ) df df x 1 x n F,..., F x n+1 x m x 1 x m ( ) M ( ) x 1 F x n ( ) ( ) x 1 F x n F F M F ) N x n+1 x m M x n+1 x m. M 1.3.8 φ(y, t) x M n+1 x m M y 1 y n fdx 1 dx m N (( ) ) (( ) df x 1 df F φ(x) x n )F ( ( N dµ M (x) M x 1 )F x n )F M φjfdµ M M ( ) φ(x)dµ f 1 (y)(x) dµ N (y) f 1 (y) N M. φjfdµ M

46 3 f Euclid x O x U x f(u x ) f(x) f : U x f(u x ) Euclid O {U x } x O O M O O {U x } x O {U k } {U k } O {U k } {ψ k } f U k f k : U k V k f(u k ) ψ k φ y (ψ k φ)(x)dµ f 1 (y)(x) (y) k V k f 1 k f 1 k V k µ N ( ) (ψ k φ)(x)dµ f 1 (y)(x) dµ Vk (y) ψ k φjfdµ Uk. (y) k U k y k f 1 k (ψ k φ)(x)dµ f 1 (y)(x) (y) k N µ N y {ψ k f 1 (y)} f 1 (y) (ψ k φ)(x)dµ f 1 (y)(x) φ(x)dµ f f 1 k (y) k f 1 (y)(x) 1 (y) k y f 1 (y) φ(x)dµ f 1 (y)(x) N µ N φjf M µ M Lebesgue φ 0 Lebesgue M φjfdµ M k k N N ψ k φjfdµ Uk U k ( ) (ψ k φ)(x)dµ f 1 (y)(x) dµ Vk (y) V f 1 k k (y) k ( ) (ψ k φ)(x)dµ f 1 (y)(x) dµ N (y) k f 1 k (y) k ( ) φ(x)dµ f 1 (y)(x) dµ N (y) f 1 (y)

3.2. 47 M φjfdµ M N ( ) φ(x)dµ f 1 (y)(x) dµ N (y) f 1 (y) 3.2.6 3.2.5 m n N y N µ N φ(x) dµ N (y) φ(x)jf(x)dµ M (x) N x f 1 (y) M x f 1 (y) 3.2.7 3.2.6 φ(x) M φjfdµ M 1.3.10 M (U; x 1,..., x n ) U φjfdµ M ( ) ( ) φ df df dx 1 dx n. x 1 x n M N e 1,..., e n x U f(x) F (x) [ ( ) ( )] df df [e 1 e n ]F (x) x 1 x n ( ) ( ) df df det F (x) x 1 x n φjfdµ M φ(x) det F (x) dx 1 dx n. U 3.2.8 (Archimedes) r > 0 S k (r) {x R k+1 x r} f : S 1 (r) ( r, r) S 2 (r) ; (r cos θ, r sin θ, t) ( r 2 t 2 cos θ, r 2 t 2 sin θ, t)

48 3 3.2.9 3.2.8 2 n ( 3.2.11) R 3 e 1, e 2, e 3 f S 1 (r) /θ ( r, r) /t S 1 (r) ( r, r) θ t (r cos θ, r sin θ, t) ( r sin θ, r cos θ, 0), θ (r cos θ, r sin θ, t) (0, 0, 1). t 1.3.7 θ t θ t r df ( ) df f θ θ ( r 2 t 2 sin θ, r 2 t 2 cos θ, 0), ( ) df f t t ( (r2 t 2 ) 1/2 t cos θ, (r 2 t 2 ) 1/2 t sin θ, 1). 1.3.7 ( ) ( ) 2 ( ) df df 2 ( ) θ t df 2 θ df t ( ) t (r 2 t 2 2 ) r 2 t + 1 r 2 2 Jf ( df ) ( θ df ) t θ t r r 1 f 3.2.10 3.2.8 f A(S 2 (r)) 1dµ 1dµ 2πr 2r 4πr 2 S 2 (r) S 1 (r) ( r,r) 2 S 2 (r)

3.2. 49 3.2.11 r > 0 D k (r) {x R k x < r} f : S 1 (r) D n 1 (r) S n (r) ; (r cos θ, r sin θ, x) ( r 2 x 2 cos θ, r 2 x 2 sin θ, x) n S 1 (r) D n 1 (r) S n (r) R n+1 R n+1 SO(2) SO(n 1) SO(2) SO(n 1) R n+1 S 1 (r) D n 1 (r) S n (r) SO(2) SO(n 1) Jf S 1 (r) D n 1 (r) SO(2) SO(n 1) (r, 0, s, 0,..., 0) (0 s < r) Jf R n+1 e 1,..., e n+1 f S 1 (r) /θ θ0 D n 1 (r) e 3,..., e n+1 S 1 (r) D n 1 (r) θ θ0 θ (r cos θ, r sin θ, s, 0,..., 0) (0, r, 0,..., 0) θ0 e 3,..., e n+1 1.3.7 θ e 3 e n+1 r e 3 e n+1 r θ0 df ( ) df θ f θ0 θ (0, r 2 s 2, 0,..., 0) r 2 s 2 e 2, θ0 df(e 3 ) f t ( r 2 (s + t) 2, 0, s + t, 0,..., 0) t0 ( (r 2 s 2 ) 1/2 s, 0, 1, 0,..., 0) (r 2 s 2 ) 1/2 se 1 + e 3 4 i n + 1 df(e i ) df dt ( i r 2 (s 2 + t 2 ), 0, s, 0,..., t,..., 0) e i. t0

50 3 1.3.7 ( ) df θ df(e 3 ) df(e n+1 ) θ0 ( ) df 2 θ df(e 3 ) 2 df(e n+1 ) 2 θ0 ( ) s (r 2 s 2 2 ) r 2 s + 1 r 2 2 ( df ) θ Jf θ0 df(e3 ) df(e n+1 ) θ0 e 3 e n+1 r r 1 f 3.2.12 ω n vol(s n 1 (1)), κ n vol(d n (1)) θ 2 vol(s n 1 (r)) ω n r n 1, vol(d n (r)) κ n r n n 2 f : D n (1) [0, 1] ; x x ( 3.2.5) Jf 1 κ n vol(d n (1)) 3.2.11 vol(d n (1)) 1 0 1 vol(s n 1 (r))dr 0 vol(s n 1 (r))dr 1 0 ω n r n 1 dr [ ωn n rn ] 1 0 ω n+1 vol(s n (1)) vol(s 1 (1))vol(D n 1 (1)) 2πκ n 1. ω 1 vol(s 0 (1)) 2, κ 1 vol(d 1 (1)) 2, ω 2 vol(s 1 (1)) 2π, κ 2 vol(d 2 (1)) π ω n κ n 3.2.13 S n 1 + {u S n 1 (1) 0 u 1 } u 1 dµ(u) vol(d n 1 (1)) κ n 1. S n 1 + ω n n.

3.2. 51 f : S+ n 1 D n 1 (1) R n {0} R n 1 R n 1 P S+ n 1 f f P f P S+ n 1 e {0} R n 1 e 1 1.3.9 Jf e, e 1 u 1 f : S+ n 1 D n 1 (1) S+ n 1 1 ( 3.2.6) S n 1 + u 1 dµ(u) S n 1 + Jfdµ(u) vol(d n 1 (1)). 3.2.14 R 2 C R 2 c : I R 2 t 0 I t s(t) dc dt dt t 0 I s(t) s(t) c c(t 0 ) c(t) t ds dt dc dt > 0 s(t) t t(s) t(s) C c(s) c(t(s)) s s d c ds dc dt dt ds dc / ds dt dt dc / dc dt dt 1 s c(s) c(s) (x(s), y(s)) e(s) c (s) (x (s), y (s)) e(s) π/2 c (s) 1 n(s) ( y (s), x (s)) c (s), c (s) 1

52 3 s 2 c (s), c (s) 0 c (s) c (s) e(s) c (s) n(s) κ(s) c (s) κ(s)n(s) κ(s) c(s) (x (s), y (s)) c (s) κ(s)n(s) κ(s)( y (s), x (s)) n (s) ( y (s), x (s)) ( κ(s)x (s), κ(s)y (s)) κ(s)e(s). e (s) κ(s)n(s), n (s) κ(s)e(s) 3.2.15 (Fenchel) c c s κ(s) 2π κ(s) ds c R 2 1 1 P 1 (R) P 1 (R) {(x, y) R 2 x cos θ + y sin θ 0} θ θ P 1 (R) dθ dθ P 1 (R) Riemann L(P 1 (R)) π c c(s) c(s) R 2 g g : c P 1 (R) C c 1 g ( 3.2.6) #(g 1 (l))dµ P 1 (R)(l) Jgdµ c P 1 (R) #X X n(s) n(s) (cos θ(s), sin θ(s)) g(s) θ(s) Jg dθ ds d (cos θ(s), sin θ(s)) ds n (s) κ(s) c

3.2. 53 P 1 (R) #(g 1 (l))dµ P 1 (R)(l) κ(s) ds. l P 1 (R) c l #(g 1 (l)) 2 κ(s) ds 2vol(P 1 (R)) 2π. c 3.2.16 3.2.15 Euclid Morse Chern-Lashof c

54 4 R 2 L(R 2 ) L(R 2 ) Riemann Riemann L(R 2 ) 4.1 4.1.1 R 2 L(R 2 ) (r, θ) L(R 2 ) l(r, θ) l(r, θ) {(x, y) R 2 x cos θ + y sin θ r} l : R 2 L(R 2 ) ; (r, θ) l(r, θ) (1) (3) (1) l (2) L(R 2 ) l Hausdorff (3) t R O t {(r, θ) R 2 r R, t < θ < t + π} U t l(o t ) φ t : U t R 2 ; l(r, θ) (r, θ) ((r, θ) O t ) (L(R 2 ), {(U 0, φ 0 ), (U π/2, φ π/2 )}) 2 (1) R 2 x cos θ + y sin θ r Hesse l (2) l l(r, θ) l(r, θ ) r r, θ θ + 2πZ r r, θ θ + π + 2πZ L(R 2 ) l 1 l 2

4.1. 55 l 1 l(r 1, θ 1 ), l 2 l(r 2, θ 2 ) r 1, θ 1, r 2, θ 2 θ 0 < θ 1, θ 2 < θ 0 + π θ 0 O R (θ 0, θ 0 + π) O R 2 O Hausdorff (r 1, θ 1 ) V 1 (r 2, θ 2 ) V 2 V 1 V 2 l(v 1 ) l(v 2 ) L(R 2 ) l 1 l 2 l(v 1 ) l(v 2 ) L(R 2 ) Hausdorff (3) U t L(R 2 ) L(R 2 ) {U t } φ t : U t φ t (U t ) φ π/2 φ 1 0 : φ 0 (U 0 U π/2 ) φ π/2 (U 0 U π/2 ) U 0 U π/2 {l(r, θ) 0 < θ < π/2 π/2 < θ < π} φ 0 (U 0 U π/2 ) {(r, θ) 0 < θ < π/2 π/2 < θ < π} φ π/2 (U 0 U π/2 ) {(r, θ) π/2 < θ < π π < θ < 3π/2}. 0 < θ < π/2 φ π/2 φ 1 0 (r, θ) φ π/2 (l(r, θ)) φ π/2 (l( r, θ + π)) ( r, θ + π). π/2 < θ < π φ π/2 φ 1 0 (r, θ) φ π/2 (l(r, θ)) (r, θ). φ π/2 φ 1 0 : φ 0 (U 0 U π/2 ) φ π/2 (U 0 U π/2 ) (U t, φ t ) t R L(R 2 ) (U t ; r t, θ t ) r t, θ t t r, θ L(R 2 ) t R U t g t dr t dr t + dθ t dθ t Riemann L(R 2 ) r t ( 1) m r t, θ t θ t + nπ (m, n ) U t U t g t Riemann g t {(U t, g t )} L(R 2 )

56 4 R 2 M(R 2 ) M(R 2 ) [ ] [ ] [ ] [ ] x 1 cos φ sin φ x 1 u 1 T (φ, u 1, u 2 ) : + sin φ cos φ x 2 φ, u 1, u 2 M(R 2 ) M(R 2 ) R 2 Lie T (φ, u 1, u 2 )l(r, θ) l(r + u 1 cos(θ φ) + u 2 sin(θ φ), θ φ) M(R 2 ) R 2 M(R 2 ) L(R 2 ) Lie T (φ, u 1, u 2 ) L(R 2 ) ( ) d(t (φ, u 1, u 2 )) r r, ( ) d(t (φ, u 1, u 2 )) ( u 1 sin(θ φ) + u 2 cos(θ φ)) θ r + θ. d(t (φ, u 1, u 2 )) ( ) ( ) d(t (φ, u 1, u 2 )) d(t (φ, u 1, u 2 )) r θ r θ d(t (φ, u 1, u 2 )) T (φ, u 1, u 2 ) L(R 2 ) 4.1.2 R 2 B D(B) {l L(R 2 ) B l } D(B) L(R 2 ) Fubini k D(B) I k (B) I k (B) D(B) [0, ] ; l L(B l) k D(B) L(B l) k dµ(l) x 2 u 2 I 1 (B) ( 4.1.4)

4.1. 57 4.1.3 R 2 B I 1 (B) πa(b) L(R 2 ) r, θ θ L(B l) r L(B l(r, θ))dr A(B) I 1 (B) L(B l)dµ(l) π D(B) 0 L(B l(r, θ))drdθ πa(b). 4.1.4 ( ) R 2 c B 4πA(B) L(c) 2 c c R 2 Riemann (0, π) R Riemann M c (0, π) Riemann 2 Riemann p : M L(R 2 ) p(c(s), φ) c(s) dc B φ ds M R ; (c(s), φ) L(B p(c(s), φ)) M 2 M M (s 1, φ 1, s 2, φ 2 ) I (L(B p(c(s 1 ), φ 1 )) sin φ 2 L(B p(c(s 2 ), φ 2 )) sin φ 1 ) 2 dµ M 2 0 M 2 I L(B p(c(s 1 ), φ 1 )) 2 sin 2 φ 2 dµ M 2 M 2 2 L(B p(c(s 1 ), φ 1 )) sin φ 1 L(B p(c(s 2 ), φ 2 )) sin φ 2 dµ M 2 M 2 + L(B p(c(s 2 ), φ 2 )) 2 sin 2 φ 1 dµ M 2 M 2 L(B p(c(s 1 ), φ 1 )) 2 dµ M sin 2 φ 2 dµ M M M 2 L(B p(c(s 1 ), φ 1 )) sin φ 1 dµ M L(B p(c(s 2 ), φ 2 )) sin φ 2 dµ M M M + L(B p(c(s 2 ), φ 2 )) 2 dµ M sin 2 φ 1 dµ M. M M

58 4 L(B p(c(s), φ)) 2 dµ M M x 2 π 0 x 0 L(c) π 0 0 L(B p(c(s), φ)) 2 dφds 2rdr L(B p(c(s), φ)) 2 dφ π L(B p(c(s),φ)) 0 0 L(B p(c(s), φ)) 2 dµ M 2L(c)A(B) M 2rdrdφ 2A(B) sin 2 φdµ M L(c) π M 0 0 sin 2 φdφds π π 0 sin 2 φdφ 1 (1 cos 2φ)dφ π 2 0 2 sin 2 φdµ M π 2 L(c) L(B p(c(s), φ)) sin φdµ M M M M L(B p(c(s), φ)) p : M L(R 2 ) 3.2.5 L(B p(c(s), φ))jpdµ M L(B p(x))dµ L(R 2 )(l) M L(R 2 ) x p 1 (l) #(p 1 (l))l(b l)dµ L(R 2 )(l). D(B) l D(B) #(p 1 (l)) 2 4.1.3 L(B p(c(s), φ))jpdµ M 2πA(B). M Jp dc ds (cos ψ(s), sin ψ(s))

4.1. 59 p(c(s), φ) ( l 0, ψ(s) + φ π ) + c(s) 2( T (0, c 1 (s), c 2 (s))l 0, ψ(s) + φ π ) ( ( 2 l c 1 (s) cos ψ(s) + φ π ) + c 2 (s) sin 2 ξ ψ(s) + φ π 2 ( ψ(s) + φ π ), ψ(s) + φ π ) 2 2 ( ) dp s ( ) dp φ { dc1 ds cos ξ + dc 2 ds sin ξ c 1(s) sin ξ dψ ds + c 2(s) cos ξ dψ } ds r + dψ ds θ, { c 1 (s) sin ξ + c 2 (s) cos ξ} r + θ. r, θ L(R2 ) 3.2.7 ( ) ( dp dp s φ) det dc 1 ds cos ξ + dc 2 ds sin ξ cos ψ(s) cos cos sin φ sin φ. dc 1 ds cos ξ + dc 2 ds sin ξ c 1 sin ξ dψ ( φ π 2 ) ( ψ(s) + φ π 2 ) dψ ds ds + c 2 cos ξ dψ ds ( + sin ψ(s) sin ψ(s) + φ π ) 2 c 1 sin ξ + c 2 cos ξ 1 L(B p(c(s), φ))jpdµ M L(B p(c(s), φ)) sin φdµ M. M M L(B p(c(s), φ)) sin φdµ M 2πA(B) M

60 4 I 2 2L(c)A(B) π 2 L(c) 2(2πA(B))2 2πA(B){L(c) 2 4πA(B)} L(c) 2 4πA(B) 0. I 0 M 2 L(B p(c(s 1 ), φ 1 )) sin φ 2 L(B p(c(s 2 ), φ 2 )) sin φ 1 0 L(B p(c(s), φ))/ sin φ M r(φ) r(φ) sin φ C (C : ) r(φ) C sin φ ( C (C sin φ cos φ, C sin 2 φ) 2 sin 2φ, C 2 (2 sin2 φ 1) + C ) 2 ( C 2 sin 2φ, C 2 cos 2φ + C ) 2 (0, C/2) C/2 4.2 Crofton L(R 2 ) 2 Crofton 4.2.1 I(R 2 L(R 2 )) {(x, l) R 2 L(R 2 ) x l} I(R 2 L(R 2 )) R 2 L(R 2 ) 3 i : R 2 P 1 (R) I(R 2 L(R 2 )) ; (x, l) (x, l + x) i i R 2 x 1, x 2 L(R 2 ) r, θ x 1, x 2, r, θ R 2 L(R 2 ) R 2 L(R 2 ) I(R 2 L(R 2 )) x 1 cos θ + x 2 sin θ r

4.2. Crofton 61 F (x 1, x 2, r, θ) x 1 cos θ + x 2 sin θ r [ F x 1 F x 2 F r ] F [cos θ sin θ 1 x 1 sin θ + x 2 cos θ] θ 1 I(R 2 L(R 2 )) R 2 L(R 2 ) 3 i i(u 1, u 2, l(0, θ)) (u 1, u 2, T (0, u 1, u 2 )l(0, θ)) (u 1, u 2, l(u 1 cos θ + u 2 sin θ, θ)) i R 2 L(R 2 ) C i I(R 2 L(R 2 )) C i i i di (u1,u 2,l(0,θ)) di (u1,u 2,l(0,θ)) di (u1,u 2,l(0,θ)) ( ) x ( 1 ) x ( 2 ) θ + cos θ x 1 r + sin θ x 2 r R 2 P 1 (R) x 1, ( u 1 sin θ + u 2 cos θ) r + θ,, x 1 x 2 θ R2 L(R 2 ), x 2 r, i di θ 1 0 0 0 1 0 cos θ sin θ u 1 sin θ + u 2 cos θ. 0 0 1 3 i 4.2.2 (Crofton ) R 2 c #(c l)dµ(l) 2L(c) L(R 2 )

62 4 c Crofton c c c [0, L(c)] {0} R 2 c 1 l(r, θ) c r, θ Crofton π/2 θ π/2 θ r 0 r L(c) cos θ π/2 L(c) cos θ π/2 0 1drdθ π/2 π/2 L(c) cos θdθ 2L(c). c x Crofton x L(R 2 ) Crofton c c 1,..., c k ( k ) k #(c l)dµ(l) # c i l dµ(l) #(c i l)dµ(l) L(R 2 ) L(R 2 ) k i1 L(R 2 ) i1 ( k ) # c i l i1 #(c i l)dµ(l) L(R 2 ) i1 k 2L(c i ) 2L(c). i1 k #(c i l) l c i l c i c i l 1 L(R 2 ) 0 ( k ) k # c i l dµ(l) #(c i l)dµ(l) L(R 2 ) i1 i1 L(R 2 ) i1 c c i d i lim L(d i ) L(c) l i #(d 2 i l) #(d 2 i+1) #(c l)

4.2. Crofton 63 c l #(d 2 i l) #(c l) c l L(R 2 ) 0 Lebesgue #(c l)dµ(l) lim #(d 2 i l)dµ(l) lim 2L(d 2 i) 2L(c) L(R 2 ) i L(R 2 ) i I(c) {(x, l) I(R 2 L(R 2 )) x c} 4.2.1 i i 1 (I(c)) c P 1 (R) R 2 P 1 (R) 2 I(c) I(R 2 L(R 2 )) 2 π L : I(c) L(R 2 ) ; (x, l) l π L π L I(c) R 2 L(R 2 ) R 2 L(R 2 ) L(R 2 ) C I(c) 1 π L 3.2.5 #(π 1 L (l))dµ(l) Jπ L dµ L(R 2 ) I(c) π 1 L (l) {(x, l) x l, x c} (c l) {l} #(π 1 L (l)) #(c l) #(c l)dµ(l) L(R 2 ) I(c) Jπ L dµ. π L dπ L c s P 1 (R) l(0, θ) θ s, θ c P 1 (R) 4.2.1 i c P 1 (R) s dc 1 + dc 2 ds x 1 ds x 2

64 4 ( ) di s ( ) di θ dc 1 ds + dc 2 x 1 ds ( dc1 + x 2 ds cos θ + dc ) 2 ds sin θ r ( c 1 (s) sin θ + c 2 (s) cos θ) r + θ I(c) ( ) ( dc1 dπ L di s ds cos θ + dc ) 2 ds sin θ r ( ) dπ L di ( c 1 (s) sin θ + c 2 (s) cos θ) θ r + θ. r, θ L(R2 ) 3.2.7 ( ) ( ) dπ Ldi dπ L di s θ dc 1 det ds cos θ + dc 2 ds sin θ c 1(s) sin θ + c 2 (s) cos θ 0 1 dc 1 ds cos θ + dc 2 ds sin θ I(c) Jπ L dµ dc ds c dc ds π 0 dc 1 ds cos θ + dc 2 ds sin θ dθds. (cos ϕ, sin ϕ) ϕ π dc 1 ds cos θ + dc 2 ds sin θ π dθ cos ϕ cos θ + sin ϕ sin θ dθ 0 0 π 0 cos(θ ϕ) dθ 2 ( s ). Fubini Jπ L dµ 2ds 2L(c) I(c) c

4.3. 65 #(c l)dµ(l) 2L(c) L(R 2 ) 4.2.3 c R 2 c (cos θ, sin θ) d c (θ) 4.2.4 4.2.2 c c B I 0 (B) π 0 d c (θ)dθ L(c) c d L(c) πd c 4.1.2 D(B) {l L(R 2 ) B l } D(B) L(R 2 ) D(B) c 2 c c L(R 2 ) 0 4.2.2 π L(c) dµ drdθ d c (θ)dθ D(B) D(B) I 0 (B) c d L(c) πd 0 4.3 R 2 M(R 2 ) M(R 2 ) [ ] [ ] [ ] [ ] x 1 cos φ sin φ x 1 u 1 T (φ, u 1, u 2 ) : + sin φ cos φ x 2 φ, u 1, u 2 M(R 2 ) [ ] T (ψ, v 1, v 2 )T (φ, u 1, u 2 ) ([ cos φ T (ψ, v 1, v 2 ) sin φ x 1 x 2 ] [ sin φ cos φ x 1 x 2 ] [ + u 1 u 2 ]) x 2 u 2

66 4 [ [ [ cos ψ sin ψ ] ([ ] [ ] [ ]) [ sin ψ cos φ sin φ x 1 u 1 v 1 + + cos ψ sin φ cos φ x 2 u 2 v 2 ] [ ] [ ] [ ] x 1 cos ψ sin ψ u 1 + x 2 sin ψ cos ψ u 2 ] [ ] [ x 1 u 1 cos ψ u 2 sin ψ + v 1 + x 2 u 1 sin ψ + u 2 cos ψ + v 2 cos(ψ + φ) sin(ψ + φ) sin(ψ + φ) cos(ψ + φ) cos(ψ + φ) sin(ψ + φ) sin(ψ + φ) cos(ψ + φ) ] + ]. [ v 1 v 2 ] M(R 2 ) R 2 x 1 cos φ sin φ u 1 T (φ, u 1, u 2 ) x 2 sin φ cos φ u 2 1 0 0 1 x 1 x 2 1. T (ψ, v 1, v 2 )T (φ, u 1, u 2 ) cos ψ sin ψ v 1 sin ψ cos ψ v 2 0 0 1 cos φ sin φ u 1 sin φ cos φ u 2 0 0 1 cos(ψ + φ) sin(ψ + φ) u 1 cos ψ u 2 sin ψ + v 1 sin(ψ + φ) cos(ψ + φ) u 1 sin ψ + u 2 cos ψ + v 2 0 0 1 T (ψ + φ, u 1 cos ψ u 2 sin ψ + v 1, u 1 sin ψ + u 2 cos ψ + v 2 ). M(R 2 ) R 2 Lie SO(2) R 2 M(R 2 ) ; (T (φ, 0), u 1, u 2 ) T (φ, u 1, u 2 ) φ, u 1, u 2 M(R 2 ) φ SO(2) Riemann R 2 Riemann M(R 2 ) 4.3.1 φ, u 1, M(R 2 ) M(R 2 ) M(R 2 ) 1 φ, u 1, u 2 φ, u 1, u 2 u 2

4.3. 67 M(R 2 ) Riemann M(R 2 ) g L g (x) gx (x M(R 2 )) L g g T (ψ, v 1, v 2 ) L g (T (φ, u 1, u 2 )) T (ψ + φ, u 1 cos ψ u 2 sin ψ + v 1, u 1 sin ψ + u 2 cos ψ + v 2 ) ( ) (dl g ) e φ e ( ) (dl g ) e u 1 e ( ) (dl g ) e u 2 e φ g cos ψ u 1 + sin ψ g u 2 g sin ψ u 1 + cos ψ g u 2 g (dl g ) e : T e (M(R 2 )) T g (M(R 2 )) L g L g 4.3.2 M(R 2 ) g R g (x) xg (x M(R 2 )) R g g T (ψ, v 1, v 2 ) R g (T (φ, u 1, u 2 )) T (φ + ψ, v 1 cos φ v 2 sin φ + u 1, v 1 sin φ + v 2 cos φ + u 2 ) ( ) (dr g ) e φ e φ + ( v 1 sin φ v 2 cos φ) g ( ) (dr g ) e u 1 e u 1 g ( ) (dr g ) e u 2 e u 2 g u 1 + (v 1 cos φ v 2 sin φ) g u 2 g (dr g ) e : T e (M(R 2 )) T g (M(R 2 )) ( ) ( ) ( ) (dr g ) e φ (dr g ) e e u 1 (dr g ) e e u 2 e φ g u 1 g u 2 4.3.1 R g R g M(R 2 ) g

68 4 4.4 Poincaré c 0, c 1 c 1 M(R 2 ) g #(c 0 cg 1 ) M(R 2 ) 4L(c 0 )L(c 1 ) Poincaré 4.4.1 I((R 2 ) 2 M(R 2 )) {(x, y, g) (R 2 ) 2 M(R 2 ) x g(y)} I((R 2 ) 2 M(R 2 )) (R 2 ) 2 M(R 2 ) 5 i : (R 2 ) 2 SO(2) I((R 2 ) 2 M(R 2 )) ; (x, y, T (φ, 0)) (x, y, T (0, x)t (φ, 0)T (0, y)) i i (R 2 ) 2 x 1, x 2 y 1, y 2 M(R 2 ) φ, u 1, u 2 x 1, x 2, y 1, y 2 φ, u 1, u 2 (R 2 ) 2 M(R 2 ) (R 2 ) 2 M(R 2 ) I((R 2 ) 2 M(R 2 )) [ ] [ ] [ ] [ ] x 1 cos φ sin φ y 1 u 1 + sin φ cos φ x 2 y 2 u 2 x 1 y 1 cos φ y 2 sin φ + u 1 x 2 y 1 sin φ + y 2 cos φ + u 2 F (x 1, x 2, y 1, y 2, φ, u 1, u 2 ) (y 1 cos φ y 2 sin φ + u 1 x 1, y 1 sin φ + y 2 cos φ + u 2 x 2 ) [ F1 x 1 F 1 x 2 F 1 y 1 F 1 y 2 F 1 φ F 2 F 2 F 2 F 2 F 2 x 1 x 2 y 1 y 2 φ [ F 1 F 1 u 1 u 2 F 2 F 2 u 1 u 2 1 0 cos φ sin φ y 1 sin φ + y 2 cos φ 1 0 0 1 sin φ cos φ y 1 cos φ y 2 sin φ 0 1 ] ]

4.4. Poincaré 69 2 I((R 2 ) 2 M(R 2 )) (R 2 ) 2 M(R 2 ) 5 i T (0, x 1, x 2 )T (φ, 0, 0)T (0, y 1, y 2 ) 1 0 x 1 cos φ sin φ 0 0 1 x 2 sin φ cos φ 0 0 0 1 0 0 1 cos φ sin φ x 1 sin φ cos φ x 2 0 0 1 1 0 y 1 0 1 y 2 0 0 1 cos φ sin φ y 1 cos φ + y 2 sin φ + x 1 sin φ cos φ y 1 sin φ y 2 cos φ + x 2 0 0 1 1 0 y 1 0 1 y 2 0 0 1 T (φ, y 1 cos φ + y 2 sin φ + x 1, y 1 sin φ y 2 cos φ + x 2 ) i(x 1, x 2, y 1, y 2, T (φ, 0)) (x 1, x 2, y 1, y 2, T (φ, y 1 cos φ + y 2 sin φ + x 1, y 1 sin φ y 2 cos φ + x 2 )) i (R 2 ) 2 M(R 2 ) C i I((R 2 ) 2 M(R 2 )) C i i i di (x1,x 2,y 1,y 2,φ) di (x1,x 2,y 1,y 2,φ) di (x1,x 2,y 1,y 2,φ) di (x1,x 2,y 1,y 2,φ) di (x1,x 2,y 1,y 2,φ) ( ) x ( 1 ) x ( 2 ) y ( 1 ) y ( 2 ) φ +, x 1 u 1 +, x 2 u 2 y 1 cos φ u 1 sin φ u 2, y 2 + sin φ u 1 cos φ u 2, φ + (y 1 sin φ + y 2 cos φ) u 1 +( y 1 cos φ + y 2 sin φ) u 2 (R 2 ) 2 SO(2),,, x 1 x 2 y 1 M(R 2 ) x 1, x 2, y 1, y 2, φ,, u 1, y 2 φ (R2 ) 2 i u 2

70 4 di 1 0 0 0 0 0 1 0 0 0 0 0 1 0 0 0 0 0 1 0. 0 0 0 0 1 1 0 cos φ sin φ y 1 sin φ + y 2 cos φ 0 1 sin φ cos φ y 1 cos φ + y 2 sin φ 5 i 4.4.2 (Poincaré ) R 2 c 0, c 1 #(c 0 gc 1 )dµ(g) 4L(c 0 )L(c 1 ) M(R 2 ) I(c 0, c 1 ) {(x, y, g) I((R 2 ) 2 M(R 2 )) x c 0, y c 1 } 4.4.1 i i 1 (I(c 0, c 1 )) c 0 c 1 SO(2) SO(2) (R 2 ) 2 3 I(c 0, c 1 ) I((R 2 ) 2 M(R 2 )) 3 π M : I(c 0, c 1 ) M(R 2 ) ; (x, y, g) g π M π M I(c 0, c 1 ) (R 2 ) 2 M(R 2 ) (R 2 ) 2 M(R 2 ) M(R 2 ) C π M 3.2.5 #(π 1 M (g))dµ(g) Jπ M dµ M(R 2 ) I(c 0,c 1 ) π 1 M (g) {(g(y), y, g)) g(y) c 0 gc 1 } #(π 1 M (g)) #(c 0 gc 1 ) #(c 0 gc 1 )dµ(g) M(R 2 ) I(c 0,c 1 ) Jπ M dµ.

4.4. Poincaré 71 π M dπ M c 0 s c 1 t SO(2) T (φ, 0) φ s, t, φ c 0 c 1 SO(2) 4.4.1 i c 0 c 1 SO(2) I(c 0, c 1 ) ( ) ( ) ( ) di, di, di s t φ I(c 0, c 1 ) dπ M Jπ M 4.4.1 di ( ) ( d(c0 (s)) 1 dπ M di dπ M di + d(c ) 0(s)) 2 s ds x 1 ds x 2 d(c 0(s)) 1 + d(c 0(s)) 2, ds u 1 ds u ( ) ( 2 d(c1 (t)) 1 dπ M di dπ M di + d(c ) 1(t)) 2 t dt y 1 dt y ( 2 d(c 1(t)) 1 cos φ + d(c ) 1(t)) 2 sin φ dt dt u ( 1 + d(c 1(t)) 1 sin φ d(c ) 1(t)) 2 cos φ, dt dt u ( ) 2 dπ M di ((c 1 (t)) 1 sin φ + (c 1 (t)) 2 cos φ) φ u 1 +( (c 1 (t)) 1 cos φ + (c 1 (t)) 2 sin φ) + u 2 φ 4.3.1 ( ) ( ) ( dπ Mdi dπ M di dπ M di s t φ) d(c 0 (s)) 1 d(c 1(t)) 1 cos φ + d(c 1(t)) 2 sin φ (c ds dt dt 1 (t)) 1 sin φ + (c 1 (t)) 2 cos φ d(c det 0 (s)) 2 d(c 1(t)) 1 sin φ d(c 1(t)) 2 cos φ (c ds dt ds 1 (t)) 1 cos φ + (c 1 (t)) 2 sin φ 0 0 1 ( d(c 0 (s)) 1 d(c 1(t)) 1 sin φ d(c ) 1(t)) 2 cos φ ds dt dt d(c ( 0(s)) 2 d(c 1(t)) 1 cos φ + d(c 1(t)) 2 sin φ). ds dt dt ( d(c0 (s)) 1, d(c ) 0(s)) 2 (cos α, sin α), ds ds ( d(c1 (t)) 1 dt, d(c ) 1(t)) 2 (cos β, sin β) dt

72 4 cos α( cos β sin φ sin β cos φ) sin α( cos β cos φ + sin β sin φ) cos α sin(φ + β) + sin α cos(φ + β) sin(φ + β α) 3.2.7 L(c0 ) L(c1 ) 2π Jπ M dµ sin(φ + β α) dφdtds I(c 0,c 1 ) 0 0 0 ( φ α, β ) L(c0 ) L(c1 ) 2π 0 0 L(c0 ) L(c1 ) 0 0 4L(c 0 )L(c 1 ). 0 4dtds sin φ dφdtds M(R 2 ) #(c 0 gc 1 )dµ(g) 4L(c 0 )L(c 1 ) 4.4.3 R 2 x r S(x; r) R 2 c R 2 #(c S(x; r))dµ(x) 4rL(c) 4.4.2 c 0 c c 1 S(0; r) #(c gs(0; r))dµ(g) 4 L(c)2πr M(R 2 ) 4.3.1 #(c gs(0; r))dµ(g) #(c S(g0; r))dµ(g) M(R 2 ) M(R 2 ) 2π #(c S(x; r))dµ(x). R 2 #(c S(x; r))dµ(x) 4rL(c) R 2

4.4. Poincaré 73 4.4.4 R 2 c 0, c 1 g M(R 2 ) c 0 gc 1 c 0 gc 1 θ i c 0 gc 1 θ i 0 θ i π c 0 gc 1 θ i dµ(g) 2πL(c 0 )L(c 1 ). M(R 2 ) c 0 gc 1 4.4.2 I(c 0, c 1 ) (x, y, g) c 0 gc 1 θ(x, y, g) I(c 0, c 1 ) θ θ 4.4.2 π M : I(c 0, c 1 ) M(R 2 ) 3.2.5 θdµ(g) θjπ M dµ M(R 2 ) π 1 M (g) I(c 0,c 1 ) M(R 2 ) c 0 gc 1 θ i dµ(g) 4.4.2 θjπ M dµ I(c 0,c 1 ) L(c0 ) L(c1 ) φ+β απ 0 0 φ+β α π φ + β α sin(φ + β α) dφdtds ( φ α, β ) L(c0 ) L(c1 ) π π π 0 0 π π φ sin φ dφ 2 0 φ sin φ dφdtds. π φ sin φdφ 2 2[ φ cos φ] π 0 + 2 2π. 0 π 0 φ( cos φ) dφ cos φdφ θjπ M dµ 2πL(c 0 )L(c 1 ) I(c 0,c 1 ) M(R 2 ) c 0 gc 1 θ i dµ(g) 2πL(c 0 )L(c 1 )

74 4 4.5 Steiner Hotelling 4.5.1 3.2.15 R 2 1 P 1 (R) Riemann dθ dθ l P 1 (R) R 2 l P l : R 2 l I [0, 1] R 1 R 2 L(P l (I))dµ(l) 2. P 1 (R) 0 θ π l(θ) {(x, y) R 2 x cos θ + y sin θ 0} P 1 (R) L(P l(θ) (I)) sin θ L(P l (I))dµ(l) P 1 (R) π 0 sin θdθ 2. 4.5.2 R 2 K W1 2 (K) W1 2 (K) 1 L(P l (K))dµ(l). 2 4.5.1 W 2 1 (I) 1 P 1 (R) 4.5.3 4.5.2 L(P l (K)) 4.2.3 l K Barbier 4.2.4 4.5.4 (Barbier ) R 2 K L(P l (K))dµ(l) 2W1 2 (K) L(K). P 1 (R) l P 1 (R) f (P l ) K : K l ( 3.2.5) K y int(p l (K)) #(f 1 (y)) 2 ( 3.2.5) Jfdµ #(f 1 (y))dµ(y) 2L(P l (K)) K P l (K) df K l K e l u 1.3.9 Jf e, u e, u dµ 2L(P l (K)). K

4.5. Steiner Hotelling 75 L(P l (I))dµ(l) 1 ( ) e, u dµ dµ(l) P 1 (R) 2 P 1 (R) K 1 ( ) e, u dµ(l) dµ 2 K P 1 (R) ( 4.5.1 ) 1 2dµ L(K). 2 K 4.5.5 (Steiner ) R 2 K ρ K ρ A(K ρ ) A(K) + L(K)ρ + πρ 2. r 0 l P 1 (R) P l (K r ) l P l (K) r L(P l (K r )) L(P l (K)) + 2r l P 1 (R) L(P l (K r ))dµ(l) L(P l (K))dµ(l) + 2πr. P 1 (R) P 1 (R) Barbier ( 4.5.4) L(K r ) L(K) + 2πr A(K ρ ) A(K) + ρ 0 L(K r )dr A(K) + A(K) + L(K)ρ + πρ 2. ρ 0 (L(K) + 2πr)dr 4.5.6 (Steiner ) R 2 K ρ K ρ ρ A(K ρ ) A(K) + L(K)ρ + πρ 2. K s K K u K e K κ(s) Frenet-Serret du ds κe, de ds κu. ρ 0 f : K [0, ρ] K ρ intk ; (x, t) x te

76 4 f ( ) df f s s u tde ds (1 + κt)u, df ( ) f t t e 1.3.7 ( ) ( ) ( ( Jf df df s t s) df t) df 1 + κt. ρ f Jf 1 + κt ( 3.2.5) 2π A(K ρ intk) Jfdµ (1 + κt)dµ K [0,ρ] K [0,ρ] ρ L(K)ρ + κds tdt L(K)ρ + 2π 1 2 ρ2 K L(K)ρ + πρ 2. A(K ρ ) A(K) + L(K)ρ + πρ 2. 0 ( 4.1.4) x R 2 r 0 B(x; r) {y R 2 x y r} M i (r) {x R 2 B(x; r) B} M e (r) {x R 2 B B(x; r)} M i (r) M e (r) r r M i (r) M i (r ), M e (r) M e (r ) I i {r R r 0, M i (r) } I e {r R r 0, M e (r) } I i I e r I i M i (r), r I e M e (r)

4.5. Steiner Hotelling 77 r i sup I i, r e inf I e M i (r i ) r I i M i (r), M e (r e ) I i [0, r i ], I e [r e, ) r I e M e (r) r i r r e r c 4.4.3 R 2 #(c S(x; r))dµ(x) 4rL(c) x #(c S(x; r)) µ R 2 D k {x R 2 #(c S(x; r)) k} D k µ R 2 c c A(D 2k 1 ) A(D ) 0. Lebesgue R 2 #(c S(x; r))dµ(x) 2kA(D 2k ). k1 ka(d 2k ) 2rL(c). k1 r B r {x R 2 c S(x; r) } A(D 2k ) A(B r ) k1 2rL(c) A(B r ) (k 1)A(D 2k ). k1

78 4 4.5.5 A(B r ) A(B) + rl(c) + πr 2 2rL(c) A(B r ) 2rL(c) (A(B) + rl(c) + πr 2 ) r ( ) 2 ( ) L(c) L(c) 2 2rL(c) A(B r ) π 2π r + 4π A(B). 2rL(c) A(B r ) L(c) 2 4π ( ) 2 L(c) A(B) π 2π r + (k 1)A(D 2k ) 0 k1 4πA(B) L(c) 2 L(c) 2π r 0 r r i r r e L(c) 2π r i r e c 4.5.7 (Bonnesen ) R 2 c B B r i B r e L(c) 2 4πA(B) π 2 (r e r i ) 2 r i r r e r L(c) 2 4π ( ) 2 L(c) A(B) π 2π r. ( ) L(c) 2 2 L(c) 4π A(B) π 2π r i ( L(c) 2 4π A(B) π L(c) ) 2 2π + r e.

4.5. Steiner Hotelling 79 1 2 (u2 + v 2 ) ( u + v 2 ) 2 ( ) L(c) 2 2 4π A(B) π re r i. 2 L(c) 2 4πA(B) π 2 (r e r i ) 2 4.5.8 (Hotelling ) R 2 c ρ c ρ ρ A(c ρ ) L(c)L( D 1 (ρ)) 2L(c)ρ c s c u u π/2 e c κ(s) Frenet- Serret du ds κe, de ds κu. ρ 0 f : c [ ρ, ρ] c ρ ; (x, t) x te f ( ) df f s s u tde ds (1 + κt)u, df ( ) f t t e 1.3.7 ( ) ( ) ( ( Jf df df s t s) df t) df 1 + κt. ρ f Jf 1 + κt ( 3.2.5) ρ A(c ρ ) Jfdµ (1 + κt)dµ 1dµ + κ(s)ds tdt c [ ρ,ρ] c [ ρ,ρ] c [ ρ,ρ] c ρ L(c)L([ ρ, ρ]) L(c)L( D 1 (ρ)) 2L(c)ρ. A(c ρ ) L(c)L( D 1 (ρ)) 2L(c)ρ.

80 4 4.6 Blaschke 4.6.1 (Blaschke ) R 2 c 0, c 1 D 0, D 1 R 2 D χ(d) χ(d 0 gd 1 )dµ(g) 2π(A(D 0 ) + A(D 1 )) + L(c 0 )L(c 1 ). M(R 2 ) 2π 2π D 0 gd 1 (D 0 gd 1 ) κ((d 0 gd 1 )) θ i ((D 0 gd 1 )) θ i ((D 0 gd 1 )) 0 κ((d 0 gd 1 ))ds + θ i ((D 0 gd 1 )) 2πχ(D 0 gd 1 ) (D 0 gd 1 ) i g M(R 2 ) θ i ((D 0 gd 1 )) g M(R 2 ) 4.4.4 θ i ((D 0 gd 1 ))dµ(g) M(R 2 ) i i M(R 2 ) c 0 gc 1 θ i dµ(g) 2πL(c 0 )L(c 1 ). (D 0 gd 1 ) c 0 gd 1 gc 1 D 0 ( ) κ((d 0 gd 1 ))ds dµ(g) M(R 2 ) (D 0 gd 1 ) c 0 gd 1 I(c 0, D 1 ) {(x, y, g) c 0 D 1 M(R 2 ) x gy} I(c 0, D 1 ) I((R 2 ) 2 M(R 2 )) i : (R 2 ) 2 SO(2) I((R 2 ) 2 M(R 2 )) i 1 (I(c 0, D 1 )) c 0 D 1 SO(2) (R 2 ) 2 SO(2) 4 π M (x, y, g) g ((x, y, g) I(c 0, D 1 ))

4.6. Blaschke 81 π M : I(c 0, D 1 ) M(R 2 ) c 0 κ 0 c 0 D 1 SO(2) κ 0 f π M i ( 3.2.5) ( ) ( ) κ 0 dµ dµ(g) κ 0 Jfdµ M(R 2 ) f 1 (g) c 0 D 1 SO(2) g SO(2) g 0 f 1 (g) {(x, y, g 0 ) x c 0, y D 1, x gy} c 0 gd 1 2 κ 0 dµ 2 κ 0 (s)ds f 1 (g) c 0 gd 1 ( ) ( ) 2 κ 0 (s)ds dµ(g) M(R 2 ) c 0 gd 1 ( ) Jf s d(c 0(s)) 1 ds + d(c 0(s)) 2 x 1 ds x 2, y 1, y 2, c 0 D 1 SO(2) df Jf 4.4.1 di ( ) ( d(c0 (s)) 1 df dπ M di + d(c ) 0(s)) 2 s ds x 1 ds x 2 d(c 0(s)) 1 + d(c 0(s)) 2, ds u 1 ds u ( ) 2 df cos φ sin φ, y 1 u 1 u 2 ( ) df y ( 2 ) df φ sin φ u 1 cos φ u 2, φ (y 1 sin φ + y 2 cos φ) u 1 + ( y 1 cos φ + y 2 sin φ) u 2 + φ df [ ] df s y 1 y 2 φ [ u 1 u 2 ] (c 0 (s)) 1 cos φ sin φ y ds 1 sin φ + y 2 cos φ (c 0 (s)) 2 sin φ cos φ y φ ds 1 cos φ + y 2 sin φ 0 0 0 1.

82 4 df 3 1 [ [ (c 0 (s)) 1 ds cos ψ, cos ψ cos φ sin φ sin ψ sin φ cos φ (c 0 (s)) 2 ds sin ψ ] 1 cos(ψ φ) sin(ψ φ) cos ψ cos φ cos(ψ φ) + sin φ sin(ψ φ) sin ψ sin φ cos(ψ φ) cos φ sin(ψ φ) ( ker df R + cos(ψ φ) + sin(ψ φ) ). s y 1 y 2 (ker df) span R { cos ψ s cos φ y 1 + sin φ y 2, sin ψ s sin φ y 1 cos φ y 2, ] [ 0 0 ] }. φ (ker df) df 1.3.8 Jf ( df cos ψ s cos φ + sin φ ) y 1 y 2 [ ] (c 0 (s)) 1 cos ψ cos φ sin φ y ds 1 sin φ + y 2 cos φ (c 0 (s)) 2 cos φ sin φ cos φ y u 1 u 2 φ ds 1 cos φ + y 2 sin φ sin φ 0 0 0 1 0 [ u 1 u 2 [ u 1 u 2 ] cos 2 ψ + cos 2 φ + sin 2 φ cos ψ sin ψ + sin φ cos φ cos φ sin φ φ 0 ] cos 2 ψ + 1 cos ψ sin ψ φ 0 (cos 2 ψ + 1) + cos ψ sin ψ, u 1 u ( 2 df sin ψ s sin φ cos φ ) y 1 y 2 [ u 1 u 2 φ ] (c 0 (s)) 1 cos φ sin φ y ds 1 sin φ + y 2 cos φ (c 0 (s)) 2 sin φ cos φ y ds 1 cos φ + y 2 sin φ 0 0 0 1 sin ψ sin φ cos φ 0

4.6. Blaschke 83 [ u 1 u 2 [ u 1 u 2 ] cos ψ sin ψ + cos φ sin φ sin φ cos φ sin 2 ψ + sin 2 φ + cos 2 φ φ 0 ] cos ψ sin ψ sin 2 ψ + 1 φ 0 cos ψ sin ψ + (sin 2 ψ + 1), u 1 u ( ) 2 df φ (y 1 sin φ + y 2 cos φ) u 1 + ( y 1 cos φ + y 2 sin φ) u 2 + φ. ( cos ψ s cos φ + sin φ ) ( sin ψ y 1 y 2 s sin φ cos φ ) y 1 y 2 φ ( cos ψ s cos φ + sin φ ) ( sin ψ y 1 y 2 s sin φ cos φ ) y 1 y 2 ( cos ψ sin φ + cos φ sin ψ) s + ( cos ψ cos φ sin φ sin ψ) y 1 s y 2 +(cos 2 φ + sin 2 φ) y 1 y 2 sin(ψ φ) s cos(ψ φ) y 1 s + y 2 y 1 y 2 sin(ψ φ) 2 + cos(ψ φ) 2 + 1 2, ( df cos ψ s cos φ + sin φ ) df y 1 y ( 2 df φ) { (cos 2 ψ + 1) + cos ψ sin ψ } u 1 u { 2 (y 1 sin φ + y 2 cos φ) + ( y 1 cos φ + y 2 sin φ) + u 1 u 2 φ} cos 2 ψ + 1 cos ψ sin ψ y 1 sin φ + y 2 cos φ det cos ψ sin ψ sin 2 ψ + 1 y 1 cos φ + y 2 sin φ 0 0 1 (cos 2 ψ + 1)(sin 2 ψ + 1) cos 2 ψ sin 2 ψ ( sin ψ s sin φ cos φ ) y 1 y 2 { cos ψ sin ψ + (sin 2 ψ + 1) } u 1 u 2

84 4 cos 2 ψ + sin 2 ψ + 1 2. 1.3.8 Jf 2 2 2 ( ) 2 κ 0 dµ 2 κ 0 (s)ds A(D 1 ) 2π c 0 D 1 SO(2) c 0 2(2π) 2 A(D 1 ). ( ) κ 0 (s)ds dµ(g) (2π) 2 A(D 1 ). M(R 2 ) c 0 gd 1 gc 1 D 0 ( ) ( ) κ 1 (s)ds dµ(g) κ 1 (s)ds dµ(g) M(R 2 ) D 0 gc 1 M(R 2 ) c 1 g 1 D ( 0 ) κ 1 (s)ds dµ(g) M(R 2 ) c 1 gd 0 (2π) 2 A(D 0 ). ( ) κ((d 0 gd 1 ))ds dµ(g) (2π) 2 (A(D 0 ) + A(D 1 )) M(R 2 ) (D 0 gd 1 ) 2π χ(d 0 gd 1 )dµ(g) M(R 2 ) ( κ((d 0 gd 1 ))ds + M(R 2 ) (D 0 gd 1 ) i (2π) 2 (A(D 0 ) + A(D 1 )) + 2πL(c 0 )L(c 1 ) θ i ((D 0 gd 1 )) ) dµ(g) χ(d 0 gd 1 )dµ(g) 2π(A(D 0 ) + A(D 1 )) + L(c 0 )L(c 1 ). M(R 2 )

85 5 Euclid 5.1 Euclid 5.1.1 R n L n 1 (R n ) (r, u) R S n 1 (1) L n 1 (R n ) l(r, u) l(r, u) {x R n x, u r} l : R S n 1 (1) L n 1 (R n ) ; (r, u) l(r, u) (1) (3) (1) l (2) L n 1 (R n ) l Hausdorff (3) v S n 1 (1) O v {(r, u) R S n 1 (1) r R, 0 < u, v } U v l(o v ) φ v : U v R n ; l(r, u) (r, u u, v v) ((r, u) O v ) (L n 1 (R n ), {(U v, φ v ) v S n 1 (1)}) n (1) R n u r x, u r l (2) l l(r, u) l(r, u ) r r, u u r r, u u L n 1 (R n ) l 1 l 2 l 1 l(r 1, u 1 ), l 2 l(r 2, u 2 ) r 1, u 1, r 2, u 2 0 < u 1, u 0, u 2, u 0 u 0 O u0 R S n 1 (1) O u0

86 5 Euclid Hausdorff (r 1, u 1 ) V 1 (r 2, u 2 ) V 2 V 1 V 2 l(v 1 ) l(v 2 ) L n 1 (R n ) l 1 l 2 l(v 1 ) l(v 2 ) L n 1 (R n ) Hausdorff (3) U v L n 1 (R n ) L n 1 (R n ) {U v } φ v : U v φ v (U v ) φ v φ 1 v : φ v (U v U v ) φ v (U v U v ) (U v, φ v ) v S n 1 L n 1 (R n ) (U v ; r v, u v ) r v, u v v r, u l : R S n 1 (1) L n 1 (R n ) ɛ(r, u) ( r, u) ((r, u) R S n 1 (1)) R S n 1 (1) Riemann R S n 1 (1) ɛ R S n 1 (1) R S n 1 (1) Riemann l : R S n 1 (1) L n 1 (R n ) L n 1 (R n ) Riemann v S n 1 (1) U v Riemann g v dr v dr v + g S n 1 (1) R n M(R n ) M(R n ) n Φ SO(n) U R n T (Φ, U) : X ΦX + U M(R n ) SO(n) R n M(R n ) R n Lie l(r, u) L n 1 (R n ) T (Φ, U)x, Φu Φx, Φu + U, Φu x, u + U, Φu r + U, Φu T (Φ, U)l(r, u) l(r + U, Φu, Φu) M(R n ) R n M(R n ) L n 1 (R n ) Lie T (Φ, u) L n 1 (R n ) u S n 1 (1) e 1,..., e n 1 l : R S n 1 (1) L n 1 (R n ) dl R S n 1 (1) L n 1 (R n ) ( ) dl, dl(e 1 ),..., dl(e n 1 ) r