202.. : r = r(u, v) u v, dv = 0, = 0, = ; E dv =. ( k gu = Γ 2 k gv = Γ 22 ( dv ) 3 E F E F 2 = Γ 2 2 E E, ) 3 E F 2 = Γ 22 E F 2., F = 0 E F k gu = Γ 2 2 E E = 2EF u EE v + F E u E F 2 2(E F 2 ) E E = E v 2E = 2 ln E v, E F k gv = Γ 2 22 = 2F v u + F v E F 2 2(E F 2 ) = u 2 E = 2 E. :, Liouville, 2 ln E cos θ + v 2 sin θ, E
u -, θ = 0, k gu = 2 ln E v, v -, θ = π 2, k gv = dθ 2 E. 2. r(u, v) = {a cos u cos v, a cos u sin v, a sin u} k g = sin u dv, θ. E = a 2, F = 0, = a 2 cos 2 u. u -, θ r u, Liouville 2 ln E cos θ + v 2 sin θ, E ln E v = 0, 2 sin u = cos u, sin θ sin u, a cos u dv = sin θ = sin θ a cos u, sin udv. 3. R a (a < R). R k n = ± R, a k = a, k2 g = k 2 k 2 n k g = ± a 2 R 2 = ± R ar 2 a 2. R, a k g = ± ar R 2 a 2. 2
, R,.,.,. 4. r = {u cos v, u sin v, av} r = {u 0 cos v, u 0 sin v, av} ( u 0 = ). I = 2 + (u 2 + a 2 )dv 2, F = 0 u v, v k gv = u 2 E = u (u 2 + a 2 ). x = u 0 cos v, y = u 0 sin v, z = av v, k gv u=u0 = u 0 u 2 0 + a2. 5. S C, C P S. C P S C C P. 6. S : r = r(u, v) (C) : u = u(t), v = v(t), t (C). k g. 7.,. xoz x = f(v), C : z = g(v), z r(u, v) = {f(v) cos u, f(v) sin u, g(v)}, 3
u -, v -,, r u = { f sin u, f cos u, 0}, r v = {f cos u, f sin u, g }; I = 2 = f 2 2 + (f ) 2 + (g ) 2 ]dv 2, F = 0,, Liouville, v - k gv = 2 E = 0,.. u - k gu = 2 ln E v f = f (f ) 2 + (g ). 2, f = 0,., P 25 254. 8. : (),. (2),. (), k g = 0;, k n = 0, k 2 = k 2 g + k 2 n k = 0,. (2) Γ τ g, Frenet τ g = dβ γ = β(α β) = ( β, α, β), α Γ, β, γ. Γ, k g = ±k sin θ = 0, θ β. k = 0,. k 0, sin θ = 0, θ = 0 θ = π, Γ β n. β = ±n. ( ) ( dn dn τ g =, α, n = dr ) n ( dn = E F 2 dr ) (r u r v ) ( ) ( ) dn dr = E F 2 r u r v 4 ( dn r v ) ( )] dr r u.
dr = r u + r dv v, τ g = E F 2 dn = n u + n dv v, ( dv )2 dv ( )2 E F L M N Γ,, 0. τ g = 0,. Γ p Σ p, : Γ Γ. 9. I = v( 2 + dv 2 ). uv., E = = v, F = 0, E v = v =, E u = u = 0. F = 0, = E cos θ = v cos θ, dv = sin θ = v sin θ, dθ E cos θ u sin θ) = 3 = 2 ( Ev E sin θ = cos θdv, dv 2v, = cot θ. () dθ =, (2) 2v sin θdθ sin θdθ = 0 (), (2) tan θdθ = dv 2v, cos θ = C v ( C ). dv = C v C 2. 5
= Cdv v C 2, u = 2C v C 2, u 2 = 4C 2 (v C 2 ). uv. 0. r = {u cos v, u sin v, av}. E =, F = 0, = u 2 + a 2. F = 0, dv E = tan θ = tan θ () u 2 + a2 (2) dθ = 2 = 2 E ln E v d ln(u 2 + a 2 ) 2 tan θ. cot θdθ = 2 d ln(u2 + a 2 ), tan θ ln sin θ = 2 ln(u2 + a 2 ) + ln C, sin θ = C u 2 + a 2, (2) cos θ = u 2 + a 2 C 2 u 2 + a 2, () dv = C u 2 + a 2 C 2 u 2 + a 2, v = C u u 0 u 2 + a 2 C 2 u 2 + a 2 6
.. Liouville : () ; (2)., I = 2 + dv 2 dv = tan θ dθ = 0 θ = const., v = u tan θ + c (c ),. r(u, v) = {a cos u, a sin u, v}, I = a 2 + dv 2 = dū 2 + d v 2 (ū = au, v = v)., v = ū tan θ + c v = (a tan θ)u + c, c, θ. r(u) = {a cos u, a sin u, (a tan θ)u + c},,, ( θ = π 2 ) ( θ = 0 ). 2. :,. 8 (2) τ g = E F 2 ( dv )2 dv ( )2 E F L M N : τ g = 0 0,. 3. : 2 = 2 + dv 2, : k g = d tan ( dv ] ) + dv. 7
, E =, F = 0, = (u, v), Liouville 2 ln E cos θ + v 2 E sin θ = dθ + 2 sin θ = dθ + dv 2 = dθ + dv. d tan ( ] dv ) = = + d = 4. d tan ( ] E tan θ) cos 2 θ + tan 2 θ = dθ. dv tan ( ] dv ) dθ + dv, k g = d tan ( dv ] ) + dv. 2 = 2 + (u, v)dv 2, u - α, dα dv =., E =, F = 0, = (u, v). u α, r u α, θ = α, E =, F = 0, = (u, v) Liouville k g = dα + dv, 8
, k g = 0, dα =. 5. :,. θ, 0 < θ π 2. u -, v -. Liouville, u -, k gu = 2, Liouville, ln E v = 0, () 2 ln E v cos θ + 2 E sin θ = 0, (2), θ, () (2) 2 E sin θ = 0, 0 < θ π 2, 2 E = 0, ( ) u E = 0, ( ) u = 0, ( u - ) K = ( ) uu, K = 0,.,, θ (0 < θ π 2 ).. u -, a, Liouville, k gu = 2 9 ln E v = a, (3)
b, Liouville, 2 ln E v cos θ + 2 E sin θ = b, (4), θ, (3) (4) a cos θ + 2 E sin θ = b, 0 < θ π 2, sin θ > 0, c = b a cos θ sin θ 2 E ( ), (4) = c, (5) (3) (5), ( ] E) v ( ] ) u E v u = a( E) v, = c( ) u,,, { K = ( ] E) v + E v ( ] ) u E = E a( E) v + c( ) u ] = (a 2 + c 2 )., a = c = 0, K 0, ; a, c 0, K,. 6. R. R r(θ, ϕ) = {R cos θ cos ϕ, R cos θ sin ϕ, R sin θ}. E = R 2 cos 2 θ, F = 0, = R 2 ; L = R cos 2 θ, M = 0, N = R. u } 0
K = LN M 2 E F 2 = R2 cos 2 θ R 4 cos 2 θ = R 2. auss-bonnet, α + α 2 + α 3 = π + Kdσ T = π + σ R 2. T, σ. 7. auss-bonnet : S,. 8. S, S. S T Γ, auss-bonnet Kdσ + k g = 2π, T = Γ, k g T = 0. Kdσ = 2π, T T T, S, auss,,,. 9. a, b (C), f (C). : () D(a + b) = Da + Db; (2) D(fa) = dfa + fda; (3) D(a b) = Da b + a Db. r, r 2, a = a i r i, b = b j r j, i, j 2,
a + b = (a i + b i )r i, i =, 2 () D(a + b) = D( (a i + b i )r i ) = d(a k + b k ) + Γ k ij(a i + b i ) j ]r k = (da k + Γ k ija i j )r k + (db k + Γ k ija i j )r k = Da + Db (2) D(fa) = D( fa i r i ) = (d(fa k ) + Γ k ij(fa i ) j )r k = ((df)a k + fda k + fγ k ija i j )r k = df( a k r k ) + f (da k + Γ k ija i j )r k = dfa + fda. (3) Da b + a Db = (da i + Γ i kl ak l )r i ( b j r j ) + (db j + Γ j pqb p q )r j ( a i r i ) = g ij (da i b j ) + g ij Γ i kl ak b j l + g ij (db j a i ) + g ij Γ j pqa i b p q = g ij d(a i b j ) + (g kj Γ k il + g ikγ k jl )ai b j l = g ij d(a i b j ) + (Γ ilj + Γ jli )a i b j l = g ij d(a i b j ) + g ij l l a i b j = g ij d(a i b j ) + (dg ij )a i b j = d( g ij a i b j ) = d(a b). r i r j = g ij. 2
20. a(s), b(s) (C) : r = r(s). a b =, - ( Levi-Civita ),. a(s), b(s) (C), Da = 0, Db = 0, d(a b) = Da b + a Db d(a b) = 0, a b =, Levi-Civita,. 3