# Differential Geometry of Submanifolds: Proceedings of the by Atsushi Kasue (auth.), K. Kenmotsu (eds.)

By Atsushi Kasue (auth.), K. Kenmotsu (eds.)

Read Online or Download Differential Geometry of Submanifolds: Proceedings of the Conference held at Kyoto, January 23–25, 1984 PDF

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Additional resources for Differential Geometry of Submanifolds: Proceedings of the Conference held at Kyoto, January 23–25, 1984

Example text

X ' irreducible immersion of the degree of an second fundamental x C M, we put = ( x E M 2 " x d > I such that If ¢(M') [16]). is full, of @ is This converse with degree the 2 4. (G,K) and the d e g r e e second degree is a i m p o r t a n t Theorem ~ second ). For x E X d that @ degree of x E M' (cf. is t o t a l l y the p a r a l l e l second There fundamental the for any fundamental are m a n y form geodesic. is not ~. The form. [ 9 ]). be a s y m m e t r i c to space with be a G - e q u i v a r i a n t 2 if and only if the symmetric immersion.

M Sp(4)/Sp(2)×Sp(2), the second Let be one of the following SU(8)/Sp(4), standard minimal : and SO(p+3)/SO(p)×S0(3) immersion of M (p~3,p#6). Then is rigid. On the other hand by the similar argument to the proof of Theorem A we have the following. Proposition : qp,q(~) 15. E6/F4 symmetric R-spaces and U(n)/0(n). is rigid as a minimal isometric immersion. Finally we ask the following Problem. : For any other compact of exceptional type, is the first irreducible symmetric standard minimal space M immersion of M rigid ?

Z2+I/ZI+~2+I) I/2, and given by 3 G into a unit P,q ~ of each G is given P,q T and m(2)+l = (P+q) (3,3) (resp. \$) , where Proposion of S0(p+o)/S0(p)xSO(q) = 20 (resp. = 20 (resp. family immersion (p,q) ~ ((3,3),(4,3),(5,3),(6,3),(4,4)), ¢2 is equivalent to (2) In case of (p,q) = m(2)+l immersion minimal minimal immersion of is nonrigid, the first standard (resp. SO(8)/S(O(4)xO(4))) (3) show that the first eigenspace S0(8)/S0(4)xS0(4)) min- is rigid. of in the space of all real- valued C~functions decomposes into two irreducible irreducible S0(8)-) modules over ~.

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