Riemannian Manifold

by allenlu2007

在討論 Information Geometry 之前,可能要先了解 Riemannian Geometry 和 Manifold.

前文說明 Manifold 在 local 表現像 Euclidean geometry.  但在 global 則不然。

所謂 Riemannian geometry, 一言以蔽之,就是曲率 (curvature) 是 manifold 內稟特性 (intrinsic features). 

和嵌入的 space 或座標無關 (invariant)。甚至在一些 non-strech non-compress 改變。某一些 curvature 是不變的,例如 A到B 的線長 Gaussian curvature.  例如一張紙的 Gaussian curvature 為 0.  

Curvature 基本上和 2nd order derivative 相關。In general 是一個 tensor (Riemann Metric).  

相較之下,斜率 (tangent) 或是切面 (affine), 1st order derivative, 只是輔助了解 manifold 的特性。並非 intrinsic or invariant.

 

Manifold 的定義: 可以參考本文 https://www.youtube.com/watch?v=zIjBArHTPZ4&list=PLwFQT9k5arPUpsFc-O-C5YfgUi8Wew1IT&index=1

有點複雜。基於 topological space, continuous, open-set Housdop, homomorphsim to affine space.

You know why I am not discussing it here.  基本觀念就是 local 表現像 Euclidean space.  Global 則不然。 

 

Riemannian Manifold 的定義: (So what is manifold first? need to define separately)

In differential geometry, a (smooth) Riemannian manifold (M) is a real smooth manifold M equipped with an inner product g_{p} on the tangent space T_{p}M at each point p that varies smoothly from point to point in the sense that if X and Y are vector fields on M, then p\mapsto g_{p}(X(p),Y(p)) is a smooth function. The family g_{p}of inner products is called a Riemannian metric (tensor).

白話文:  黎曼流形包含一個 M (流形) 和一個 g tensor (Riemann metric).  M 是在一個 tangent space TpM 中,每一點 p 都有對應的 g (i.e. gp) tensor.  且 gp 是平滑改變。 gp 必須是 positive semi-definite 才是 Riemann metric. 

如果 gp = I (identity matrix) everywhere, 就是 Euclidean space.  何謂 local 都是 Euclidean geometry?   想像在地球上每一點都有 gp ≠ 1.  但在夠小的 surface, 歐氏幾何仍然適用。

從 Riemannian tensor 可以定義相關的幾何特性 (on this manifold, of course!) 如角度, 線長, 面積和體積, curvature(曲率), gradient (梯度), and divergence (散度) of vector fields.  除了 divergence 之外,其他都是熟悉的幾何特性。  

Riemannian tensor or metric tensor 其實和 arclength 是等價的定義。事實上 Gauss 在証明”絕妙定理”時是用 arclength, 而非 metric tensor 來証明。Riemann 推廣到高維同時引入 metric tensor.

 

From Wiki “Metric tensor”

Arclength

If the variables u and v are taken to depend on a third variable, t, taking values in an interval [ab], then \scriptstyle{\vec{r}(u(t),v(t))} will trace out a parametric curve in parametric surface M. The arclength of that curve is given by the integral

 \begin{align}
s &= \int_a^b\left\|\frac{d}{dt}\vec{r}(u(t),v(t))\right\|\,dt \\
&= \int_a^b \sqrt{u'(t)^2\,\vec{r}_u\cdot\vec{r}_u + 2u'(t)v'(t)\, \vec{r}_u\cdot\vec{r}_v+ v'(t)^2\,\vec{r}_v\cdot\vec{r}_v}\,\,\, dt ,
\end{align}

where  \left\| \cdot \right\|  represents the Euclidean norm. Here the chain rule has been applied, and the subscripts denote partial derivatives (\scriptstyle \vec{r}_u=\tfrac{\partial \vec{r}}{\partial u}\scriptstyle \vec{r}_v=\tfrac{\partial \vec{r}}{\partial v}). The integrand is the restriction[1] to the curve of the square root of the (quadraticdifferential

ds^2 = E \,du^2 + 2F \,du\, dv + G\, dv^2 ,

 

 

 

 

 

(1)

where

 E=\vec r_u\cdot\vec r_u, \quad
F=\vec r_u\cdot\vec r_v , \quad
G=\vec r_v\cdot \vec r_v .

 

 

 

 

 

(2)

The quantity ds in (1) is called the line element, while ds2 is called the first fundamental form of M. Intuitively, it represents the principal part of the square of the displacement undergone by \scriptstyle{\vec{r}(u,v)} when u is increased by du units, and v is increased by dv units.

Using matrix notation, the first fundamental form becomes


\begin{align}
ds^2 &=
\begin{bmatrix}
du&dv
\end{bmatrix}
\begin{bmatrix}
E&F\\
F&G
\end{bmatrix}
\begin{bmatrix}
du\\dv
\end{bmatrix}\\
\end{align}

如前所述,如果 [E F; F G] 是 PSD,  則稱為 Riemannian metric.  因此 Riemannian geometry 是非歐幾何。一般稱為 elliptic geometry.

另外 arclength 也是 manifold M 的 intrinsic feature.  不會隨著座標變換而改變。 

Length and Angle

Gauss 所用的是測量平面上的線長和夾角。M 上任一點的 tangent vector 可以表示為

 \mathbf{p} = p_1\vec{r}_u + p_2\vec{r}_v

for suitable real numbers p1 and p2. If two tangent vectors are given

\mathbf{a} = a_1\vec{r}_u + a_2\vec{r}_v
 \mathbf{b} = b_1\vec{r}_u + b_2\vec{r}_v

then using the bilinearity of the dot product,


\begin{align}
\mathbf{a} \cdot \mathbf{b} &= a_1 b_1 \vec{r}_u\cdot\vec{r}_u + a_1b_2 \vec{r}_u\cdot\vec{r}_v + b_1a_2 \vec{r}_v\cdot\vec{r}_u + a_2 b_2 \vec{r}_v\cdot\vec{r}_v\\
&= a_1 b_1 E + a_1b_2 F + b_1a_2 F + a_2b_2G \\
&=\begin{bmatrix}
a_1 & a_2
\end{bmatrix}
\begin{bmatrix}
E&F\\F&G
\end{bmatrix}
\begin{bmatrix}
b_1 \\ b_2
\end{bmatrix}
\end{align}.

另一個寫法是

g(a, b) = a1*b1 E + (a1*b2 + a2*b1) F + a2*b2 G.   如果 F= 0 就是一般 Euclidean space 的內積。

g(a, b) = g(b, a)

It is also bilinear, meaning that it is linear in each variable a and b separately. That is,

g(\lambda\mathbf{a}+\mu\mathbf{a'},\mathbf{b}) = \lambda g(\mathbf{a},\mathbf{b}) + \mu g(\mathbf{a'},\mathbf{b}),\quad\text{and}
g(\mathbf{a}, \lambda\mathbf{b}+\mu\mathbf{b'}) = \lambda g(\mathbf{a},\mathbf{b}) + \mu g(\mathbf{a},\mathbf{b'})

for any vectors aa′, b, and b′ in the uv plane, and any real numbers μ and λ.

In particular, the length of a tangent vector a is given by

 \left\| \mathbf{a} \right\| = \sqrt{g(\mathbf{a},\mathbf{a})}

and the angle θ between two vectors a and b is calculated by

 \cos\theta = \frac{g(\mathbf{a},\mathbf{b})}{ \left\| \mathbf{a} \right\| \, \left\| \mathbf{b} \right\| } .

 

Area

The surface area is another numerical quantity which should depend only on the surface itself, and not on how it is parameterized. If the surface M is parameterized by the function \vec{r}(u,v) over the domain D in the uv-plane, then the surface area of M is given by the integral

\iint_D \left|\vec{r}_u\times\vec{r}_v\right|\,du\,dv

where × denotes the cross product, and the absolute value denotes the length of a vector in Euclidean space. ByLagrange’s identity for the cross product, the integral can be written

\begin{align}
\iint_D &\sqrt{(\vec{r}_u\cdot\vec{r}_u)(\vec{r}_v\cdot\vec{r}_v)-(\vec{r}_u\cdot\vec{r}_v)^2}\,du\,dv\\
&\quad=\iint_D\sqrt{EG-F^2}\,du\,dv\\
&\quad=\iint_D\sqrt{\operatorname{det}\begin{bmatrix}E&F\\ F&G\end{bmatrix}}
\, du\, dv\end{align}

where det is the determinant.

Det 代表 area 應該沒有什麼意外。因為在線代已有同樣的結果。唯一的差別是用 tangent vector space 的 local coordinate.

 

 

Geodesic 

Levi .. connection





 

Ricci Flow and Poincare Conjecture (now Theorem)

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