L n+1 n ± r L n+1 L n+1 n v v = (0, 0, , 0, 1) ∈ L n+1 n ± r n ± r n ± r n ± r n ± r n ± r n L n+1 R n+1 x, y = n k=1 x k y k − x n+1 y n+1 , x = (x 1 , x 2 , . . . , x n+1 ), y = (y 1 , y 2 , . . . , y n+1 ) ∈ R n+1 L n+1 x ∈ L n+1 x ||x|| = |x, x|. x ∈ L n+1 x = 0 x x, x > 0 x, x < 0 x, x = 0 x, y ∈ L n+1 x, y = 0. Π m L n+1 Π m Π 0 Π m Π Π m Π Π m L n+1 Π m m m HP (q, c) = {x ∈ L n+1 | x, q = c} q c L n+1 HP (q, c) q n n H n (−1) H n (−1) = {x ∈ L n+1 | x, x = −1}. n H n + (−1) H n + (−1) = {x ∈ L n+1 | x, x = −1, x n+1 > 0}. n a ∈ L n+1 r ∈ R + H n + (a, r) H n + (a, r) = {x ∈ L n+1 | x − a, x − a = −r, x n+1 ≥ 0}. n HP (q, c) n H n + (−1) n HP (q, c) ∩ H n + (−1) HP (q, c) HP (q, c) n H n−1 + (a, r) a r H n−1 + (a, r) {x n+1 = c} c H(a, r) SH(a, r, c) = H(a, r) ∩ {x n+1 = c} n (n − 1) n ± r L n+1 n ± r n ± r M = X(U ) L n+1 M L n+1 g M g p (w 1 , w 2 ) = w 1 , w 2 , ∀w 1 , w 2 ∈ T p M, ∀p ∈ M. M T p M p ∈ M M p N p M N p M = N ∈ L n+1 | N, X u i (p) = 0, i = 1, 2, . . . , n − 1 . M p ∈ M T p M (n −1) N p M n ± r M n v 1 H n + (v, 1) = {x ∈ R n+1 | x − v, x − v = −1, x n+1 ≥ 0}, v = (0, 0, . . . , 0, −1) ∈ L n+1 H n + (v, 1) n x n+1 ([5]) Π r > 0 {x = (x 1 , x 2 , . . . , x n+1 ) ∈ Π ∩ H n + (v, 1) | x n+1 = r} M L n+1 p ∈ M {x n+1 = r}, (r > 0) N p M ∩ H n + (v, 1) n ± r (p) n ± r (p) det(X u 1 , X u 2 , . . . , X u n−1 , n + r (p), n − r (p)) > 0. HS r + (v, 1) = H n + (v, 1) ∩ {x n+1 = r}, r > 0 n ± r : M → HS r + (v, 1) p → n ± r (p) n ± r M L n+1 p = X(u 1 , u 2 , . . . , u n−1 ) M n ± r (p) X u i , n = 0, i = 1, 2, . . . , n − 1, n − v, n − v = −1, n n+1 = r > 0. N n ± r p M = N p M ∩ T n ± r (p) H n + (v, 1) dn ± r p : T p M → T n ± r (p) H n + (v, 1) = T p M ⊕ N n ± r p M. π n ± r (p) T : T p M ⊕ N n ± r p M → T p M, π n ± r (p) N : T p M ⊕ N n ± r p M → N n ± r p M ⊂ N p M, A n ± r p : T p M → T p M A n ± r p = −π n ± r (p) T ◦ dn ± r p A n ± r p n ± r M p k n ± r 1 (p), k n ± r 2 (p), . . . , k n ± r n−1 (p) A n ± r p n ± r M p n ± r M p K n ± r p n ± r K n ± r p = det(A n ± r p ). n ± r π n ± r (p) N = 0, ∀p dn ± r p ≡ A n ± r p . M g ij = X u i , X u j , i = 1, 2, . . . , n −1. M p ∈ M b n ± r ij (p) = ∂ 2 X ∂u i ∂u j (p), n ± r (p), i, j = 1, 2, . . . , n −1. p M L n+1 n ± r T p M n ± r k n ± r i (p), i = 1, 2, . . . , n − 1 M p k det(b n ± r ij (p) − kg ij (p)) = 0; n ± r K n ± r p K n ± r p = k n ± r 1 (p).k n ± r 2 (p) . . . k n ± r n−1 (p) = det(b n ± r ij (p)) det(g ij (p)) . n + r n + r SH(a, r, c) M p ∈ M n + r n + r A n + r p = k n + r p id T p M k n + r p = k n + r 1 (p) = k n + r 2 (p) = ··· = k n + r n−1 (p) n + r M p r n − r M n + r n − r M n + r n − r r M n ± r M n + r n − r M H + H − p ∈ M r p p n + r p n − r p M H ± M H + H − M H M n + r n − r r M n + r n − r M H X : (0, π 2 ) × (− π 2 , 0) → L 4 ; (u, v) → (u, sin v, v, cos u). M A n + r M k n + r 1 = r cos u ; k n + r 2 = sin v r 2 cos 2 u + 2r (1 + cos 2 v) 3 . p = X(u, v) ∈ M r p = 2 sin 2 v (1 + cos 2 v) 3 − cos 2 u sin 2 v p n + r p r p p p ∈ M M H n ± r M H M n ± r n + r U ⊂ R n−1 M = X(U ) r > 0 M n + r [(n + r ) T u i ] u j = [(n + r ) T u j ] u i n + r k n + r 1 = k n + r 2 = ··· = k n + r n−1 = k n + r n − r M n + r p ∈ M −(n + r ) T ◦ dn + r | p = k n + r (p)id T p M , −(n + r ) T ◦ dn + r | p (X u i (p)) = k n + r (p)X u i (p), i = 1, 2, . . . , n −1 −(n + r ) T u i (u 1 , u 2 , . . . , u n+1 ) = k n + r (p).X u i (p), i = 1, 2, . . . , n −1. u j −(n + r ) T u i = k n + r X u i (n + r ) T [−(n + r ) T u i ] u j = k n + r u j .X u i + k n + r .X u i u j . −[(n + r ) T u j ] u i = k n + r u i .X u i + k n + r .X u j u i . −[(n + r ) T u j ] u i = −[(n + r ) T u i ] u j X u i u j = X u j u i k n + r u i X u j − k n + r u j X u i = 0. {X u i , X u j } k n + r u i = k n + r u j = 0 p ∈ M U k n + r U M n + r n + r k n + r M L 4 M = X(R 2 ) X(u, v) = (0, u, v, u 2 + v 2 + 1 − 1); (u, v) ∈ R 2 . H 3 + (v, 1) {x 1 = 0} A n ± r M k n + r p = −r √ u 2 + v 2 + 1 M M n + r M n + r n + r n + r M M n + r n + r (n + r ) u i = kX u i , i = 1, 2, , n (n + r ) T u i = (n + r ) u i , i = 1, 2, , n k M = HP (q, c) ∩ H n + (v, 1) M n v 1 M = HP (q, c) ∩ H n + (v, 1) p ∈ M M p X : U → L n+1 X −v, X − v = −1 ⇒ X u i , X −v = 0, i = 1, 2, . . . , n − 1 X u i , i = 1, 2, . . . , n − 1 M Y = X − v {Y, q} Y q {Y, q} q {Y, q} m ∈ R Y = mq X, q = c ⇒ Y + v, q = c ⇒ mq, q = c − q n+1 . q q, q = 0 m M {Y, q} M X u i , Y = 0, X u i , q = 0, i = 1, 2, . . . , n − 1, Y, q ∈ N p M {Y, q} N p M n n ± r n ∈ N p M n = λY + µq r > 0 n n − v, n − v = −1, n n+1 = r, ⇔ − λ 2 + 2λµ(c − q n+1 ) + µ 2 q, q − 2r = 0, λ = − q n+1 y n+1 µ + r y n+1 . λ, µ y n+1 Y, Y = −1 y n+1 = 0 ϕ 1 , ϕ 2 , ψ 1 , ψ 2 : U → R (λ 1 , µ 1 ) = (ϕ 1 (u 1 , u 2 , . . . , u n−1 ), ψ 1 (u 1 , u 2 , . . . , u n−1 )), (λ 2 , µ 2 ) = (ϕ 2 (u 1 , u 2 , . . . , u n−1 ), ψ 2 (u 1 , u 2 , . . . , u n−1 )). n ± r = ϕY + ψq, p = X(u 1 , u 2 , . . . , u n−1 ) ∈ M, ϕ, ψ : U → R (n ± r ) u i = ϕ u i Y + ϕY u i + ψ u i q = ϕ u i Y + ϕX u i + ψ u i q, , i = 1, 2, , n b ij (n ± r ) = n ± r , X u i u j = −(n ± r ) u j , X u i = −ϕX u i , X u j = −ϕg ij i, j = 1, 2, . . . , n −1. A n ± r p = ϕid T p M M H n + (v, 1) H n + (a, r) a ∈ R n+1 , r ∈ R + n a r M = HP (q, c) ∩H n + (v, 1) n + r n + r M M SH(a, r, c) (i) ⇒ (ii) : n + r ϕ u i Y +ψ u i q = 0, i = 1, 2, , n {Y, q} ϕ u i = ψ u i = 0, i = 1, 2, , n ϕ n + r (ii) ⇒ (iii) ϕ ϕ − ϕ 2 + 2ϕψ(c − q n+1 ) + ψ 2 q, q − 2r = 0 ϕy n+1 + ψq n+1 = r. ϕ ψ y n+1 n + 1 x n+1 M (iii) ⇒ (i) n x n+1 q = (0, . . . , 0, 1) X, q = c ⇒ x n+1 = c ⇒ y n+1 = c −1 = const ϕ y n+1 ϕ ψ M n + r M M n + r n + r M SH(a, r, c) ((i) =⇒ (ii)) : M n + r n + r λ ∈ R, λ = 0 π n + r (p) T ◦ dn + r | p = λdX| p , ∀p ∈ M dn + r = λdX ⇔ d(λX −n + r ) = 0 ⇒ ∃X 0 ∈ L n+1 : λX −n + r = X 0 . X − 1 λ (X 0 + v) = 1 λ (n + r − v) ⇒ X − 1 λ (X 0 + v), X − 1 λ (X 0 + v) = − 1 λ 2 . M 1 λ (X 0 + v) 1 λ 2 . x n+1 (u 1 , . . . , u n−1 ) = 1 λ (x 0 n+1 + r) = c = const. M SH(X 0 + v, 1 λ 2 , c) ((i) ⇐= (ii)) : M SH(a, r, c) M n ± r r n ± r H M H M L n+1 L n+1 n ± r L n+1