MINIMAL SUBMANIFOLDS IN A RIEMANNIAN MANIFOLD OF QUASI CONSTANT CURVATURE

A Riemannian manifold V~m which admits isometric imbedding into two spaces V~(m+p)ofdifferent constant curvatures is called a manifold of quasi constant curvature.TheRiemannian curvature of V~m is expressible in the formand conversely.In this paper it is proved that if M~n is any compact minimal submanifoldwithout boundary in a Riemannian manifold V~(n+p)of quasi constant curvature,then∫_(M~u)(2-1/p)σ~2-[na+1/2(b-丨b丨)(n+1)]σ+n(n-1)b~2*丨≥0,where σ is the square of the norm of the second fundamental form of M~n When V~(n+p)is amanifold of constant curvature,b=0,the above inequality reduces to that of Simons.

MINIMAL SUBMANIFOLDS IN A RIEMANNIAN MANIFOLD OF QUASI CONSTANT CURVATURE

A Riemannian manifold $[{V^m}]$ which admits isometric imbedding into two spaces $[{V^{m + p}}]$ of different constant curvatures is called a manifold of quasi constant $[{rm{curvatur}}{{rm{e}}^{[2]}}]$. The Riemannian curvature of $[{V^m}]$ is expressible in the form$$[{K_{ABCD}} = a({g_{AC}}{g_{BD}} - {g_{AD}}{g_{BC}}) + b({g_{AC}}{lambda _B}{lambda _D} + {g_{BD}}{lambda _A}{lambda _C} - {g_{AD}}{lambda _B}{lambda _C} - {g_{BC}}{lambda _A}{lambda _D}),(1 = sumlimits_{}^{} {{g_{AB}}{lambda _A}{lambda _B}} )]$$and conversely. In this paper it is proved that if $[{M^n}]$ is any compact minimal submanifold without boundary in a Riemannian manifold $[{V^{n + p}}]$ of quasi constant curvature, then$$[{{{{ left( {2 - frac{1}{p}} right){sigma ^2} - [na + frac{1}{2}(b - left| b right|)(n + 1)]sigma + n(n - 1){b^2}} }^*}| ge 0}]$$where $[sigma ]$ is the square of the norm of the second fundamental form of $[{M^n}]$. When $[{V^{n + p}}]$ is amanifold of constant curvature, $[b = 0]$, the above inequality reduces to that of Simons.