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流形积
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     THE HYPERSURFACES OF A LOCALLY PRODUCT MANIFOLD
     局部流形的超曲面
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     SUBMANIFOLDS IN A RIEMANNIAN PRODUCT MANIFOLD
     黎曼流形的子流形
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     ON THE σ-PRODUCTS
     σ-
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     Overview of manifold learning
     流形学习概述
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     On Geodesies in Kaehlerian Manifolds
     Kaehler流形上的测地线
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  manifold product
During development, a main focus has to be put on an optimum granule formulation to fulfill the manifold product and performance criteria.
      


In the first place,the paper establishes the C ∞ mapping of product space G×G/H into G/H,where G/H(G/H)is obtained from closed subgroup H(H)of Group G,and H is a corresponding Group under any C ∞ left transitive action in G/H.In the second place,the paper deals with the C ∞ mapping establised as above under particular conditions when H is the center of G,and the left action is l~.At the same time it gives some corollaries and propositions

本文首先建立由李群 G与 G的闭子群 H得到的齐性流形的积空间到 G/ H~ 的光滑映射 ,其中H~ 为 G/ H中点在左传递作用下的同位群。其次 ,在特殊情况下 ,即当 H为 G的中心 ,左作用为 l~时 ,讨论了所建立的光滑映射 ,同时又给出了几个推论及命题

The complete minimal hypersurfaces in a Locally symmetric manifold Nn+1 are studied, with some characteristics of these hypersurfaces obtained by using the generalized maximal principle. It is shown that if M is a complete minimal hypersurface in Nn+1, then M is totally geodesic or sup S is not less than (2δ-1)n. And it is shown furtuer that M is totally geodesic or M is the product of Riemannian manifold of m dimensional and n-m dimensional, whose constant sectional curvature...

The complete minimal hypersurfaces in a Locally symmetric manifold Nn+1 are studied, with some characteristics of these hypersurfaces obtained by using the generalized maximal principle. It is shown that if M is a complete minimal hypersurface in Nn+1, then M is totally geodesic or sup S is not less than (2δ-1)n. And it is shown furtuer that M is totally geodesic or M is the product of Riemannian manifold of m dimensional and n-m dimensional, whose constant sectional curvature is n/m and m/(n-m), respectively; or sup S is larger than (2δ-1)n. These results generalize the result of Shui N.X. and improve the result of Hineva S.

研究了局部对称黎曼流形Nn+1中的完备极小浸入超曲面,利用广义极大值原理给出了这种完备极小浸入超曲面全测地的特征,即若M是Nn+1中的完备极小浸入超曲面,则或者M全测地,或者M的第二基本形式模长平方的上确界supS不小于(2δ-1)n.进一步,或者M全测地,或者M是m维常数截面曲率为n/m和n-m维常数截面曲率为m/(n-m)的黎曼流形之积,或者supS大于(2δ-1)n.所得结果推广了水乃翔等关于紧致极小浸入超曲面的一个结果,并使HinevaS等人的结果成为直接推论.

 
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