Báo cáo nghiên cứu khoa học: "Tính dẹt của mặt đối chiều hai spacelike trong ln+1" pot

L n+1 n ± r L n+1 n ± r 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 ± 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. a, b ∈ L n+1 a = 0, b, b = −c < 0 a, b = 0 a, a > 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 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 ± 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 Π r > 0 {x = (x 1 , x 2 , . . . , x n+1 ) ∈ Π ∩ H n + (v, 1) | x n+1 = r} Π {a, b} a b a, b = 0 Π Π x = λa + µb. λ, µ x ∈ H n + (v, 1) x n+1 = r > 0  x − v, x − v = −1, x n+1 = r ⇔  −λ 2 + µ 2 − 2(λa n+1 + µb n+1 ) = 0, λa n+1 + µb 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 n ± r M p K n ± r p n ± r K n ± r p = det(A n ± r p ) n ± r M p H n ± r p = 1 n−1 tr(A n ± r 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 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)) . p ∈ M (u 1 , u 2 , . . . , u n−1 ) ∈ U n ± r (u 1 , u 2 , . . . , u n−1 ) = n ± r (X(u 1 , u 2 , . . . , u n−1 )). n ± r : U → H n + (v, 1). α(t) = X(u 1 (t), u 2 (t), . . . , u n−1 (t)), t ∈ (−ε, ε), ε > 0 M α(0) = p β(t) = n ± r (α(t)) = n ± r (u 1 (t), u 2 (t), . . . , u n−1 (t)), t ∈ (−ε, ε) H n + (v, 1) β(0) = n ± r (0) dn ± r  n−1  i=1 X u i (p)u  i (0)  = dn ± r (α  (0)) = d dt n ± r (u 1 , u 2 , . . . , u n−1 )   t=0 = n−1  i=1 (n ± r ) u i (0)u  i (0). dn ± r (X u i ) = (n ± r ) u i , i = 1, 2, . . . , n − 1 (n ± r ) u i , X u j  = (n ± r ) u j , X u i , i, j = 1, 2, . . . , n − 1. n ± r , X u i  = 0 u j (n ± r ) u j , X u i  = −n ± r , X u i u j . n ± r , X u j  = 0 u i (n ± r ) u i , X u j  = −n ± r , X u j u i  = n ± r , X u i u j . (n ± r ) u i , X u j  = (n ± r ) u j , X u i . dn ± r (X u i ), X u j  = dn ± r (X u j ), X u i . dn ± r = π n ± r T ◦ dn ± r + π n ± r N ◦ dn ± r π n ± r N ◦ dn ± r (X u i ), X u j  = 0, i, j = 1, 2, . . . , n − 1; π n ± r T ◦ dn ± r (X u i ), X u j  = π n ± r T ◦ dn ± r (X u j ), X u i . A n ± r p (X u i ), X u j  = A n ± r p (X u j ), X u i . {X u 1 , X u 2 , . . . , X u n−1 } T p M w 1 , w 2 ∈ T p M A n ± r p (w 1 ), w 2  = A n ± r p (w 2 ), w 1  A n ± r p (a ij ) A n ± r p {X u 1 (p), X u 2 (p), . . . , X u n−1 (p)} T p M A n ± r p (X u j (p)) = n−1  i=1 a ij X u i (p), i = 1, 2, . . . , n − 1. −dn ± r (X u j (p)), X u m (p) = A ± p (X u j (p)), X u m (p) = n−1  i=1 a ji X u i (p), X u m (p) = n−1  i=1 a ji g im (p), m = 1, n − 1, j = 1, n − 1. b n ± r jm (p) = X u j u m (p), n ± r (p) = −dn ± r (X u j (p)), X u m (p) m = 1, n − 1, j = 1, n − 1. (b n ± r jm (p)) = (a ji )(g im (p)) ⇒ (a ji ) = (b n ± r jm (p))(g im (p)) −1 , {X u 1 (p), X u 2 (p), . . . , X u n−1 (p)} det(g im (p)) = 0 (g im ) −1 . n ± r M p det[(a ij ) − kI] = 0 ⇔ det[b n ± r jm (p) − k(g ij (p))] = 0. n ± r M p (a ij ) K n ± r p = det(a ij ) = det(b n ± r p (p)) det(g ij (p)) . L n+1 M L n+1 r p ∈ M n + r n − r A n + r p = 0 A n − r p = 0 M n + r M n + r r M n ± r M n + r n − r M p ∈ M p n + r p n − r r > 0 n + r n + r M L n+1 M n + r n − r n + r (n − r ) 1. ⇒ 2. (n + r ) u i = 0, i = 1, 2, . . . , n − 1 i (n + r ) u i = 0 M n + r n + r 0 X u i u j , n + r  = 0, i, j = 1, 2, . . . , n − 1. X u j , n + r  = 0, j = 1, 2, . . . , n − 1 ⇒ X u j u i , n + r  = −X u j , (n + r ) u i , X u j , (n + r ) u i  = 0, j = 1, 2, . . . , n − 1 (n + r ) u i ∈ N p M, j = 1, 2, . . . , n − 1 λ, µ ∈ R (n + r ) u i = λn + r + µn − r , i = 1, 2, . . . , n − 1. (n + r ) n+1 = (n − r ) n+1 = r (n + r ) u i = λ(n + r − n − r ) = λ[(n + r − v) − (n − r − v)]. n + r − v, n + r − v = −1 u i (n + r ) u i , n + r − v = 0 λ(n + r − v) − (n − r − v), n + r − v = 0 ⇒ λ(−1 − n + r − v, n − r − v) = 0. (n + r ) u i = 0 λ = 0 n + r − v, n − r − v = −1. (n + r ) u i , (n + r ) u i  = λ 2 (−2 − 2n + r − v, n − r − v) = 0. (n + r ) n+1 = r (n + r ) u i , (n + r ) u i  = n  j=1 [(n + r ) j ] 2 u i = 0, (n + r ) u i = 0 n + r 2 ⇒ 1 n + r X u i , n + r  = 0, i = 1, 2, . . . , n − 1, b ij (n + r ) = X u i u j , n + r  = −X u i , (n + r ) u j  = 0, i = 1, 2, . . . , n − 1. (a ij ) A n ± r P {X u i } i=1,n−1 (a ij ) = (b ij (n + r )).(g ij ) −1 = 0. M n + r M L n+1 r > 0 n + r n − r M r > 0 n + r n − r r 1 , r 2 (r 1 = r 2 ) n + r 1 , n + r 2 n + r 1 , n − r 2 ) M (n − 1) n ± r r > 0 L 4 M n + r X, n + r  = c ∈ R ∂ ∂u i X, n + r  = X u i , n + r  + X, (n + r ) u i  = 0, i = 1, 2, . . . , n − 1. X, n + r  = c ∈ R. M M M n ± r M r > 0 n + r n − r r 1 , r 2 (r 1 = r 2 ) n + r 1 , n − r 2 n ± r M n + r n − r n + r L 4 X : (0, π 2 ) × (0, π 2 ) → L 4 (u, v) → (u, v, sin v cos u, u + v − ε), ε > 0 M = X((0, π 2 ) × (0, π 2 )) 2 L 4 r = 2 n + 2 = (2, 2, 0, 2), n − 2 = 2(1 + cos(u + v) sin v sin u, 1 − cos(u + v) cos v cos u, 1). M n + 2 n − 2 r = 1 n ± 1 = (1 + sin v sin u 2 cos(u + v) ±  cos 2 (u + v) + 2 2 , 1 − cos v cos u 2 cos(u + v) ±  cos 2 (u + v) + 2 2 , 2 cos(u + v) ±  cos 2 (u + v) + 2 2 , 1). n + 1 n − 1 M n + 1 n − 1 L n+1 n ± r L n+1