By Richard S. Palais

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**Example text**

B y assumption w e have wi. = f(x)wi. 4) 2. 13), we obtain dwi~ = df A wi + f E wij A a~j J = E fjcoj A COi "J- f E cOij A ~Oj J J J J So ~ j f j w i A ~ i = O, which implies that f j = 0 for a l l j # i. , f = c a constant. If c = O, then wi,~ = 0. , X ( M ) is contained in a hyperplane. If c 7{ O, then coi~ = cwi and d X + = E |• iei -- --03ia ¢ ei--=0. So X + e~/c is equal to a constant vector xo E R n+l, which implies that IIX - x01t 2 = ( 1 / c ) 2. 10. Definition. An immersed submanifold M n of the simply connected space form Nn+k(c) is called totally umbilic if I I = ~1, where ~ is a parallel normal field on M .

Wi,n+l = cotOwi and I I = c o t 0 I . g. in S n+l, if cos 0 = 0 (or equivalently V is a linear hyperplane). 8. E x a m p l e . and Submanifold Theory Let v0 be a non-zero vector o f the Lorentz space M= {x e R ' ~ + l ' l I ( x , x ) = - 1 , R n+l'l, (X, V o ) = a } . 2) for s o m e b. Note that (vo, vo) = --a 2 + b2. 2), we have ~-~i(awi + k~i,,~+l)ei = 0. So a w i + bcvi,n+l = 0. 3) (i) If (vo,v0) = 1, then - a 2 + b2 = 1 and we may assume that a = s i n h t o and b = cosh to. , M is totally umbilic with sectional curvature - 1 + t a n h 2 to = - s e t h 2 to.

PROOF. It follows from the definition of O3AB that w = (WAB) is an o(n + k ) - v a l u e d 1-form on M. 4) imply that w satisfies Maurer-Cartan equation: d~z7 = ~v A ~ , which is the integrability condition for the first order system d~ = w~2. 40 Part I Submanifold Theory So there exist a small neighborhood U of :Co in M and maps eA : U ~ R "+k such that deA = E a C A B @ eB, B where eA(xO) = VA and {eA(Z)} is orthonormal for all z E U. 5) is zero and hio~j = hjcd implies the second term is zero.