Let x:M→S^(n+p)(1)be an n-dimensional submanifold immersed in an(n+p)-dimensional unit sphere S^(n+p)(1).In this paper,we study n-dimensional submanifolds immersed in S^(n+p)(1)which are critical points of the functi...Let x:M→S^(n+p)(1)be an n-dimensional submanifold immersed in an(n+p)-dimensional unit sphere S^(n+p)(1).In this paper,we study n-dimensional submanifolds immersed in S^(n+p)(1)which are critical points of the functional S(x)=∫_(M)S^(n/2)dv,where S is the squared length of the second fundamental form of the immersion x.When x:M→S^(2+p)(1)is a surface in S^(2+p)(1),the functional S(x)=∫_(M)S^(n/2)dv represents double volume of image of Gaussian map.For the critical surface of S(x),we get a relationship between the integral of an extrinsic quantity of the surface and its Euler characteristic.Furthermore,we establish a rigidity theorem for the critical surface of S(x).展开更多
文摘Let x:M→S^(n+p)(1)be an n-dimensional submanifold immersed in an(n+p)-dimensional unit sphere S^(n+p)(1).In this paper,we study n-dimensional submanifolds immersed in S^(n+p)(1)which are critical points of the functional S(x)=∫_(M)S^(n/2)dv,where S is the squared length of the second fundamental form of the immersion x.When x:M→S^(2+p)(1)is a surface in S^(2+p)(1),the functional S(x)=∫_(M)S^(n/2)dv represents double volume of image of Gaussian map.For the critical surface of S(x),we get a relationship between the integral of an extrinsic quantity of the surface and its Euler characteristic.Furthermore,we establish a rigidity theorem for the critical surface of S(x).
