一类乘积空间中的子流形几何

发布时间：2018-06-21 06:15

  本文选题：乘积空间 + 平行平均曲率向量　； 参考：《大连理工大学》2015年博士论文

【摘要】：子流形几何是微分几何中的一个重要分支.近二十年来,对乘积空间中的子流形研究非常广泛,尤其是对乘积空间Mn(c) x R中的子流形的研究更加火热.本文主要研究了Mn(c) x R中的具有平行平均曲率向量场的子流形以及Willmore子流形.首先,在第三章中,我们研究了伪黎曼乘积空间Mn(c) x R中的子流形.2011年,M.Batista[1]在黎曼乘积空间M2(c)×R中具有常平均曲率的曲面上引进了一个特殊的(1,1)型张量S,并得到了关于S的一些拼挤(Pinching)常数.之后,D. Fetcu H. Rosebberg[2]把张量S推广到了一般余维数的曲面上.我们将其进一步推广到外围空间为伪黎曼乘积空间上去,并研究了算子S的间隙问题也得到了一些拼挤常数.特别地,对M2(c)×R中曲面的情况,我们得到的若干Pinching常数都优于[1]中相应的Pinching常数.其次,第四章研究了Mn(c) x R中的高斯曲率非负的曲面,并在常角条件下完全刻画了高斯曲率为零的曲面.这恰好解决了H. Alencar, M. do Carmo R. Tribuzy[3]提出的一个公开问题.我们知道要完全刻画Mn(c) x R中的平坦曲面是非常困难的,甚至对外围空间为M2(c)×R的情况都不明朗.在常角条件下,我们得到了Mn(c) x R中的平坦曲面的参数表示.再次,在第五章中我们研究了Mn(c) x R中子流形的刚性问题.通过计算一些算子的拉普拉斯,我们得到了若干个Simons型方程.从这些Simons型方程出发,我们获得了若干个间隙定理.具体来说,首先分别对Sn(1)x R中的超曲面和高余维数的子流形,我们证明了在一定条件下,子流形是Sn(1)中的全测地子流形；其二,对M3(c)×R中的曲面进行了一些分类,其中在增加额外条件下定理5.14改进了[4]中的命题4.1；其三,我们证明了Mn(c) x R中的子流形在一定条件下是Mn(c)的全测地子流形Mm+1(c)中具有常平均曲率的全脐超曲面.最后,在第六章中我们研究了Mn(c) x R中的Willmore子流形.通过计算泛函R(x)(k=n/2为Willmore泛函)的变分得到了Euler-Lagrange方程,并给出了Mn(c)xR中的子流形是Willmore子流形的充要条件.利用这些结论,我们证明了具有常角性质的Willmore曲面∑2 (?) M2(c) x R只能是∑2 (?) M2(c)和∑2=γ×R两大类(γ为M2(c)中的曲线).此外,我们还证明了全脐曲面∑2 (?) M2(c) x R必定是Willmore曲面.显然,其逆命题未必成立!为此,我们给出了使逆命题成立的一个充分条件.
[Abstract]:In this paper , we have studied the submanifolds in the product space Mn ( c ) x R . In the third chapter , we have studied the submanifolds in the product space Mn ( c ) x R . In the third chapter , we studied the submanifolds in the pseudo - Riemann product space Mn ( c ) x R . In 2011 , M . In this paper , we extend the tensor S to the surface of the general remainder . We extend the tensor S to the pseudo - Riemann product space , and study the clearance problem of the operator S . In particular , we get some Pinching constants for the surface of M2 ( c ) 脳 R . In addition , the fourth chapter studies the non - negative surface of Gaussian curvature in Mn ( c ) x R , and describes the surface with Gaussian curvature zero at constant angle . In the fifth chapter , we study the rigidity of the planar curved surface in Mn ( c ) x R . In the fifth chapter , we study the rigidity of the planar curved surface in Mn ( c ) x R . In the fifth chapter , we have studied the rigidity of Mn ( c ) x R neutron flux . In the fifth chapter , we have obtained a number of Simons equation . From these Simons equations we have obtained several gap theorems . In particular , we prove the submanifold of Sn ( 1 ) x R and the submanifold of high residual dimension respectively , and we prove that under certain conditions , the submanifold is the whole measuring submanifold in Sn ( 1 ) ;
Secondly , some classification is made on the surface of M3 ( c ) 脳 R , where Theorem 5.14 improves the proposition 4.1 in Theorem 4 .
Thirdly , we prove that the submanifolds in Mn ( c ) x R are all - umbilical hypersurfaces with constant mean curvature in the submanifold Mm + 1 ( c ) of Mn ( c ) under certain conditions . Finally , in Chapter 6 , we studied the Willmore submanifolds in Mn ( c ) x R . By calculating functional R ( x ) ( k = n / 2 , Willmore functional ) , we obtain the necessary and sufficient conditions for the submanifolds in Mn ( c ) xR . By using these conclusions we prove that the Willmore surface 鈭，

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