Questions in category: 黎曼几何 (Riemannian Geometry)

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41. [Def]完备非紧流形的端

Posted by haifeng on 2012-03-17 17:12:24 last update 2012-03-21 21:50:10 | Answers (0) | 收藏

Def1. 端

Def2. 端是正则的

$E_r:=\{x\in E : \text{dist}(x,\partial E)=r\}.$

$U_r:=\{x\in E : \frac{r}{2}<\text{dist}(x,\partial E)<2r\}$

$\sup_{E_r}u\leq C\inf_{E_r}u,$

Def3. 抛物端

(i) $f|_{\partial E}=1$;

(ii) $\liminf_{y\rightarrow E(\infty)}f(y)<1$, (其中 $E(\infty)$ 指 $E$ 的无穷远)

(i) $f|_{\partial E}=1$;

(ii) $\liminf_{y\rightarrow E(\infty)}f(y)=0$, (其中 $E(\infty)$ 指 $E$ 的无穷远)

References:

M.P.Cavalcante, H.Mirandola, and F.Vitório, The non-parabolicity of infinite volume ends. arXiv:1201.6391v1 [math.DG] 30 Jan 2012. http://arxiv.org/abs/1201.6391

Grigor’yan, A., Analytic and geometric background of recurrence and non-explosion of the
Brownian motion on Riemannian manifolds. Bull. Amer. Math. Soc. (N.S.) 36 (1999), no. 2,
135 – 249.

42. 单纯形体积(Gromov 体积)

Posted by haifeng on 2011-08-24 19:47:15 last update 2011-08-24 19:47:15 | Answers (0) | 收藏

$\|M^n\|=\inf\{\sum_i|r_i|\},$

43. Soul Theorem(Cheeger-Gromoll 灵魂定理)

Posted by haifeng on 2011-08-03 20:18:09 last update 2011-08-03 20:41:52 | Answers (0) | 收藏

References:

44. 什么样的测地线称为射线(ray)?

Posted by haifeng on 2011-08-02 18:52:46 last update 2011-08-02 18:52:46 | Answers (0) | 收藏

45. 凸集与全凸集的联系及例子

Posted by haifeng on 2011-08-02 10:37:54 last update 2011-08-02 10:51:01 | Answers (0) | 收藏

• 根据 Cartan-Hadamard 定理, 对于具有非正截面曲率的完备单连通黎曼流形, 其中的凸集一定是全凸集.
• 对于具有非正截面曲率的完备单连通黎曼流形, 它的所有的测地球(开的或闭的)都是全凸集. 特别的, 这样的流形上每点都是全凸的.
• 一般情形下, 单点集不一定是全凸集.

References:

46. 【Def】全测地子流形(totally geodesic submanifold)

Posted by haifeng on 2011-08-02 10:17:26 last update 2011-08-03 20:23:28 | Answers (0) | 收藏

47. 【Def】全凸子集(totally convex set)

Posted by haifeng on 2011-08-02 09:20:49 last update 2011-08-02 10:24:35 | Answers (0) | 收藏

• 如果对任意 $p,q\in C$, 都存在 $C$ 中最短的测地线连接 $p,q$, 并且连接 $p,q$ 的任何最短测地线都位于 $C$ 中, 则称 $C$ 是凸的. 这个概念是欧氏空间中凸集概念的推广.
• 如果上述连接任意两点 $p,q$ 的最短测地线是惟一的, 则称 $C$ 是强凸的.

48. [Gromoll, Meyer 1969]对任意维数的黎曼流形, 只要截面曲率为正, 都有simple point存在.

Posted by haifeng on 2011-08-02 08:50:20 last update 2011-08-02 08:50:20 | Answers (0) | 收藏

D. Gromoll and W. Meyer, On complete open manifolds of positive curvature, Ann. of Math. 90(1969),75-90.

49. [Cohn-Vossen]2维正截面曲率黎曼流形的 simple point 的集合总是非空.

Posted by haifeng on 2011-08-01 23:05:13 last update 2011-08-01 23:12:49 | Answers (0) | 收藏

simple point 的定义

50. 【Def】Simple point

Posted by haifeng on 2011-08-01 23:02:31 last update 2011-08-02 09:16:40 | Answers (0) | 收藏

simple point 的一个自然推广是紧致全测地子流形或紧致全凸子流形, 我们称为 $M$ 的 soul(灵魂).

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