本文選題：乘積空間 + 平行平均曲率向量��； 參考：《大連理工大學(xué)》2015年博士論文

【摘要】：子流形幾何是微分幾何中的一個重要分支.近二十年來,對乘積空間中的子流形研究非常廣泛,尤其是對乘積空間Mn(c) x R中的子流形的研究更加火熱.本文主要研究了Mn(c) x R中的具有平行平均曲率向量場的子流形以及Willmore子流形.首先,在第三章中,我們研究了偽黎曼乘積空間Mn(c) x R中的子流形.2011年,M.Batista[1]在黎曼乘積空間M2(c)×R中具有常平均曲率的曲面上引進(jìn)了一個特殊的(1,1)型張量S,并得到了關(guān)于S的一些拼擠(Pinching)常數(shù).之后,D. Fetcu H. Rosebberg[2]把張量S推廣到了一般余維數(shù)的曲面上.我們將其進(jìn)一步推廣到外圍空間為偽黎曼乘積空間上去,并研究了算子S的間隙問題也得到了一些拼擠常數(shù).特別地,對M2(c)×R中曲面的情況,我們得到的若干Pinching常數(shù)都優(yōu)于[1]中相應(yīng)的Pinching常數(shù).其次,第四章研究了Mn(c) x R中的高斯曲率非負(fù)的曲面,并在常角條件下完全刻畫了高斯曲率為零的曲面.這恰好解決了H. Alencar, M. do Carmo R. Tribuzy[3]提出的一個公開問題.我們知道要完全刻畫Mn(c) x R中的平坦曲面是非常困難的,甚至對外圍空間為M2(c)×R的情況都不明朗.在常角條件下,我們得到了Mn(c) x R中的平坦曲面的參數(shù)表示.再次,在第五章中我們研究了Mn(c) x R中子流形的剛性問題.通過計算一些算子的拉普拉斯,我們得到了若干個Simons型方程.從這些Simons型方程出發(fā),我們獲得了若干個間隙定理.具體來說,首先分別對Sn(1)x R中的超曲面和高余維數(shù)的子流形,我們證明了在一定條件下,子流形是Sn(1)中的全測地子流形；其二,對M3(c)×R中的曲面進(jìn)行了一些分類,其中在增加額外條件下定理5.14改進(jìn)了[4]中的命題4.1；其三,我們證明了Mn(c) x R中的子流形在一定條件下是Mn(c)的全測地子流形Mm+1(c)中具有常平均曲率的全臍超曲面.最后,在第六章中我們研究了Mn(c) x R中的Willmore子流形.通過計算泛函R(x)(k=n/2為Willmore泛函)的變分得到了Euler-Lagrange方程,并給出了Mn(c)xR中的子流形是Willmore子流形的充要條件.利用這些結(jié)論,我們證明了具有常角性質(zhì)的Willmore曲面∑2 (?) M2(c) x R只能是∑2 (?) M2(c)和∑2=γ×R兩大類(γ為M2(c)中的曲線).此外,我們還證明了全臍曲面∑2 (?) M2(c) x R必定是Willmore曲面.顯然,其逆命題未必成立!為此,我們給出了使逆命題成立的一個充分條件.
[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，

本文編號：2047552