The Pinching Conditions About Minimal Submanifolds Of Quasi Constant Curvature Space

Submanifolds theory is one of the main branch of differential geometry, one of the main content of study submanifolds geometry is Pinching. Pinching problem in the Euclidean space, sphere, local symmetric space and quasi constant curva-ture space and so on all have been studied. In this article mainly using a quasi constant.curvature space as outer space, using the square of the length of the sec-ond fundamental form of submanifolds, Ricci curvature infimum or codimension of quasi constant curvature space, we prove that two sufficient conditions for the com-pact minimal subinanifolds without boundary of quasi-constant curvature space is of totally geodesic submanifolds.Let Mn be a compact submanifolds which is minimally immersed in a quasi constant curvature Riemann manifold Nn+P, Q denotes Ricci curvature infimum of M. We denote by S the square of the length of h. either(1)S＜[3p-2p][na+21=3n(b-\b\-nb) or(2)S<na-4[(n-1)(a+\b\)-Q]-bn+21+3n(b-\b\): Then M is totally geodesic submanifolds. where p≥1, a. b denote smooth function and satisfy Kijkl=a{gikgjl-gilgjk)+b(gikλjλl+gjlλiλk-gilλjλk-gjkλiλl).