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《Journal of Geometry and Physics》2006,56(7):1144-1174
Let be an n-dimensional spacelike hypersurface of a constant sectional curvature Lorentz manifold . Based on previous work of S. Montiel, L. Alías, A. Brasil and G. Colares studied what can be said about the geometry of M when is a conformally stationary spacetime, with timelike conformal vector field K. For example, if has constant higher order mean curvatures and , they concluded that is totally umbilical, provided on it. If div() does not vanish on they also proved that is totally umbilical, provided it has, a priori, just one constant higher order mean curvature.In this paper, we compute for such an immersion, and use the resulting formula to study both r-maximal spacelike hypersurfaces of , as well as, in the presence of a constant higher order mean curvature, constraints on the sectional curvature of M that also suffice to guarantee the umbilicity of M. Here, by we mean the linearization of the second order differential operator associated to the r-th elementary symmetric function on the eigenvalues of the second fundamental form of x. 相似文献
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In the biharmonic submanifolds theory there is a generalized Chen’s conjecture which states that biharmonic submanifolds in a Riemannian manifold with non-positive sectional curvature must be minimal. This conjecture turned out false by a counter example of Y.L. Ou and L. Tang in Ou and Tang (2012). However it remains interesting to find out sufficient conditions which guarantee this conjecture to be true. In this note we prove that:1. Any complete biharmonic submanifold (resp. hypersurface) in a Riemannian manifold with non-positive sectional curvature (resp. Ricci curvature) which satisfies an integral condition: for some , , where is the mean curvature vector field of , must be minimal. This generalizes the recent results due to N. Nakauchi and H. Urakawa in Nakauchi and Urakawa (2013, 2011).2. Any complete biharmonic submanifold (resp. hypersurface) in a Riemannian manifold of at most polynomial volume growth whose sectional curvature (resp. Ricci curvature) is non-positive must be minimal.3. Any complete biharmonic submanifold (resp. hypersurface) in a non-positively curved manifold whose sectional curvature (resp. Ricci curvature) is smaller than for some which satisfies that is of at most polynomial growth of , must be minimal.We also consider -superbiharmonic submanifolds defined recently in Wheeler (2013) by G. Wheeler and prove similar results for -superbiharmonic submanifolds, which generalize the result in Wheeler (2013). 相似文献
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In this paper we investigate Leibniz algebras whose quotient Lie algebra is a naturally graded filiform Lie algebra . We introduce a Fock module for the algebra and provide classification of Leibniz algebras whose corresponding Lie algebra is the algebra with condition that the ideal is a Fock -module, where is the ideal generated by squares of elements from .We also consider Leibniz algebras with corresponding Lie algebra and such that the action gives rise to a minimal faithful representation of . The classification up to isomorphism of such Leibniz algebras is given for the case of . 相似文献
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Given a supervector bundle , we exhibit a parametrization of Quillen superconnections on by graded connections on the Cartan–Koszul supermanifold . The relation between the curvatures of both kind of connections, and their associated Chern classes, is discussed in detail. In particular, we find that Chern classes for graded vector bundles on split supermanifolds can be computed through the associated Quillen superconnections. 相似文献