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2004•Acta Scientiarum Naturalium Universitatis NeimongolRequires access

A Compact Submanifold of the Constant Curvature Space

JI Xing-min, Hong Wang

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Abstract

A compact submanifold M~n in the constant curvature space S~(n+p)(c) is studied.Several sufficient conditions of M~n to ensure that it is a totally geodesic (totally umbilical) submanifold are obtained.That is:let M~n be a compact minimum submanifold of constant curvature space S~(n+p)(c),if it satisfies one of the two following conditions,1) σnc(4-3n)+4nQn-4,2) K1n-4(nc-2c-Q),then M is a totally geodesic.Let M~n be a compact submanifold with parallel mean curvature vector of constant curvature space S~(n+p)(c),if it satisfies one of the two following conditions,1) σnc+2nH~2,2) Kc+H~22,then M is a totally umbilical.

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A compact submanifold M~n in the constant curvature space S~(n+p)(c) is studied.Several sufficient conditions of M~n to ensure that it is a totally geodesic (totally umbilical) submanifold are obtained.That is:let M~n be a compact minimum submanifold of constant curvature space S~(n+p)(c),if it satisfies one of the two following conditions,1) σnc(4-3n)+4nQn-4,2) K1n-4(nc-2c-Q),then M is a totally geodesic.Let M~n be a compact submanifold with parallel mean curvature vector of constant curvature space S~(n+p)(c),if it satisfies one of the two following conditions,1) σnc+2nH~2,2) Kc+H~22,then M is a totally umbilical.

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Available abstract

A compact submanifold M~n in the constant curvature space S~(n+p)(c) is studied.Several sufficient conditions of M~n to ensure that it is a totally geodesic (totally umbilical) submanifold are obtained.That is:let M~n be a compact minimum submanifold of constant curvature space S~(n+p)(c),if it satisfies one of the two following conditions,1) σnc(4-3n)+4nQn-4,2) K1n-4(nc-2c-Q),then M is a totally geodesic.Let M~n be a compact submanifold with parallel mean curvature vector of constant curvature space S~(n+p)(c),if it satisfies one of the two following conditions,1) σnc+2nH~2,2) Kc+H~22,then M is a totally umbilical.

Key concepts: Submanifold, Constant curvature, Constant (computer programming), Curvature, Mean curvature, Mathematics, Space (punctuation), Geodesic

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