→ math1209:-) 05/22 09:33
推 math1209:-) 05/22 09:36
推 Lindemann:謝謝herstein大大^^ 05/22 15:35
推 aaaaasuede:h大是Frank嗎? 05/22 21:13
→ herstein:sure... 05/23 05:48
※ 引述《Lindemann (時空之旅計時開始)》之銘言：
: 想說直接問概念比較快,我只有學和念到Gauss-Bonnet定理而已(很久以前)
: 請教一下各位大大有關Riemann curvature和Ricci curvature的問題
: 1.在古典的微分幾何,我知道Riemann curvature的推導至少有二種
: 第一個是用協變微分來算這一點我很容易理解,因為協變微分是對一個彎曲的物體
: 算一個空間的曲率必須是要繞一個封閉的路徑才能求Riemann curvature 出來
: 所以就是 Α 走一圈的變化率
: μ
: Α = Α － Γ Α
: μ;ν μ,ν μν σ
: 二次協變微分當然就是一直照定義做了再相減可以得到Riemann curvature
On R^n, denote D_i the derivative of vector fields (or functions) along
the i-th components. If the vector field V is smooth, we know
D_iD_j V - D_jD_iV=0.
On a (Riemannian) manifold, given a connection D, D_iD_j V-D_jD_iV may not
be zero. The Riemann curvature operator R measures how one changes the order
of differentiation of vector fields (or more general some tensors). If M
is a surface in R^3 with the induced metric, you can obtain the intrinsic
curvature K from the curvature operator R. Maybe it is the reason why R
is called the (Riemann) curvature operator. There are a lot of different
notions of curvatures on a Riemannian manifold when the dimension is greater
than 2. One generalizes the idea of Gaussian curvature to so-called the
sectional curvature. Moreover, if we fix two vector fields X,Y, consider
the map R_XY :Z -> R(X,Z)Y. Then this is a linear operator on the space
of smooth vector fields. The trace of this operator is called Ricci curvature
denoted by Ric(X,Y). Similarly, you can define the total curvature of the
Riemannian manifold. These notions of curvature are all intrinsic.
We don't need to embed the manifold into some Euclidean space to define
those curvatures. The mean curvature are not intrinsic. It depends on
how you embed your manifold into a bigger manifold.
For any connection D, you can still define the notion of curvature
without using the metric on it. The curvature F is defined to be D^2.
Locally, the connection D is of the form D=d+A. Here d is the De Rham
differential and A is called the connection one-form. Physicists call A
the gauge field. Well, it's very difficult to say the curvature tells you
the shape of the manifold when the connection is not Riemannian. You can
still know how a vector parallel transports along some bundles by
the given connection.
Well...I think I have to stop here....It's a long story.
Contraction: roughly you are taking trace of some linear operator.
: α
: Α － Α ＝ Α Ｒ
: μ;ν;σ μ;σ;ν α μνσ
: Ｒ有點複雜不打了,R是curvature tensor
: 這是我在物理書上看到的方法
: 但是我們在微分幾何的課有學到一個很類似的方法
: Gauss公式 ｋ ｋ
: → ^ →
: ｒ ＝Γ +ｂ ｎ 其中 ｒ 表示對ij微分
: ij ij ij ij
: Γ:Christoffel符號 ｂ:second fundamental form係數
: Weingarten公式
: ^ ｌｋ →
: ｎ ＝－ｂ ｇ ｒ ｒ表示對k微分
: j jｌ ｋ k
: 我們知道陳省身大師說到微分幾何裡面看到方程沒事兒就要給他微一微,
: 所以微分幾何就是一直微分呀XDD
: → →
: 然後再微分一次相減 ｒ － ｒ 這過程和結果懶得打了
: ijk ijk
: 就可以得到著名的Gauss-Codazzi方程,
: 就是first fundamental form和second fundamental form 的
: 係數 g_ij和 b_ij所滿足的曲面基本方程
: 這裡面Gauss方程式會自動emerge Riemann curvature出來
: 我們知道first fundamental form 和second fundamental form也可以知道曲面的形狀
: 請問一下這二件事情為何是一樣的?
: 2.什麼是Ricci tensor???
: λν
: 為何Ricci tensor定義是對curvature tensor作contraction
: λν ν
: ｇ Ｒ ＝Ｒ ＝ Ｒ
: λμνρ μνρ μρ
: 他的幾何意義為何?
: 3.Gauss-Bonnet定理曲線必須是在geodesic線上才能做嗎?
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