用户: Hui/黎曼几何

本文主要收集一些黎曼几何中常用的结论和公式.

1向量丛上的联络

考虑向量丛 $V\to M$ 上的线性联络 (cf. [EL83],[BC2010])

$$\nabla :C(TM)\times C(V)\to \mathcal{C}(V)$$(1)

定义 1.1. 内积: $$⟨\sigma \oplus \lambda ,\rho \oplus \mu ⟩=⟨\sigma ,\rho ⟩+⟨\lambda ,\mu ⟩V_x\oplus W_x$$$$⟨\sigma \otimes \lambda ,\rho \otimes \mu ⟩=⟨\sigma ,\rho ⟩\cdot ⟨\lambda ,\mu ⟩V_x\otimes W_x$$$$⟨\alpha ,\beta {⟩}_{V^{\ast }}={⟨{\alpha }^{\#},{\beta }^{\#}⟩}_V$$$$⟨\sigma ,\rho {⟩}_{{\varphi }^{-1}w}=⟨\sigma ,\rho {⟩}_b$$对偶联络: $$\left({\nabla }_X^{\ast }\theta \right)\cdot \sigma =X\cdot (\theta \cdot \sigma )-\theta \cdot {\nabla }_X\sigma$$直和: $${\nabla }_X(\sigma \oplus \lambda )={\nabla }_X^V\sigma \oplus {\nabla }_X^W\lambda$$张量积: $${\nabla }_X(\sigma \otimes \lambda )=\left({\nabla }_X^V\sigma \right)\otimes \lambda +\sigma \otimes \left({\nabla }_X^W\lambda \right)$$拉回联络: $$\begin{array}{rl}{\nabla }_X\rho & ={\nabla }_X\left(f^{\alpha }\cdot {\varphi }^{\ast }{\lambda }_{\alpha }\right)=\left(X\cdot f^{\alpha }\right){\varphi }^{\ast }{\lambda }_{\alpha }+f^{\alpha }\cdot {\nabla }_X{\varphi }^{\ast }{\lambda }_{\alpha }\\ & =\left(X\cdot f^{\alpha }\right){\varphi }^{\ast }{\lambda }_{\alpha }+f^{\alpha }\cdot {\varphi }^{\ast }\left({\nabla }_{d\varphi \cdot X}^W{\lambda }_{\alpha }\right).\end{array}$$(2)

命题 1.2. Levi-Civita 联络: $${\nabla }_X^{{\varphi }^{-1}TN}d\varphi \cdot Y-{\nabla }_Y^{{\varphi }^{-1}TN}d\varphi \cdot X=d\varphi \cdot ([X,Y])$$(3)

定义 1.3. 定义联络曲率 $R:{\bigwedge }^{2}C(TM)\otimes C(V)\to C(V)$ 为$$R(X,Y)\sigma =-{\nabla }_X{\nabla }_Y\sigma +{\nabla }_Y{\nabla }_X\sigma +{\nabla }_{[X,Y]}\sigma =-R(Y,X)\sigma$$

命题 1.4.

•	对于 $V^{\ast }:R^{\ast }(X,Y)\theta \cdot \sigma =-\theta \cdot R(X,Y)\sigma$, 其中 $X,Y\in C(TM),\theta \in C\left(V^{\ast }\right),\sigma \in C(V)$;
•	对于 $V\oplus W:R(X,Y)(\sigma \oplus \lambda )=R^V(X,Y)\sigma \oplus R^W(X,Y)\lambda$, 其中 $\lambda \in \mathcal{C}(W)$;
•	对于 $V\otimes W:R(X,Y)(\sigma \otimes \lambda )=\left(R^V(X,Y)\sigma \right)\otimes \lambda +\sigma \otimes R^W(X,Y)\lambda$;
•	对于 ${\varphi }^{-1}W:R_x(X,Y)\rho (x)=R_{\varphi (x)}^W(d\varphi \cdot X,d\varphi \cdot Y)\rho (x)$.

例 1.5 (乘积流形的曲率). $${\mathrm{Rm}}_{g_1+g_2}(X,Y,Z,W)={\mathrm{Rm}}_{g_1}\left(X_1,Y_1,Z_1,W_1\right)+{\mathrm{Rm}}_{g_2}\left(X_2,Y_2,Z_2,W_2\right)$$(4)$${\mathrm{Ric}}_{g_1+g_2}(X,Y)={\mathrm{Ric}}_{g_1}\left(X_1,Y_1\right)+{\mathrm{Rc}}_{g_2}\left(X_2,Y_2\right)$$(5)$$S=S_1+S_2.$$(6)

命题 1.6. 拉回与取对偶和张量积可交换: $${\left(f^{\ast }E\right)}^{\ast }=f^{\ast }\left(E^{\ast }\right)\text{ and }\left(f^{\ast }{E}_1\right)\otimes \left(f^{\ast }{E}_2\right)=f^{\ast }\left(E_1\otimes E_2\right)$$(7)

命题 1.7. 设 $S\in \Gamma \left(E_1^{\ast }\otimes E_2\right),\xi \in \Gamma \left(E_1\right)$ 且 $X,Y\in \mathcal{X}(M)$. 则$$(R(X,Y)S)(\xi )=R_{{\nabla }^{(2)}}(X,Y)(S(\xi ))-S\left(R_{{\nabla }^{(1)}}(X,Y)\xi \right).$$

命题 1.8 (曲率与 trace 可交换). 设 $F,G\in {\mathcal{T}}_{\ell }^k(M)$, 则$$\begin{array}{rl}R(X,Y)(\mathrm{tr}F)& =\mathrm{tr}(R(X,Y)F)\\ R(X,Y)(F\otimes G)& =(R(X,Y)F)\otimes G+F\otimes (R(X,Y)G)\end{array}$$(8)以及$$\begin{array}{rl}R(X,Y)(F({\omega }^1,\dots ,{\omega }^{\ell },Z_1,\dots ,& Z_k))=(R(X,Y)F)\left({\omega }^1,\dots ,{\omega }^{\ell },Z_1,\dots ,Z_k\right)\\ & +\sum _{j=1}^{\ell }F\left({\omega }^1,\dots ,R(X,Y){\omega }^j,\dots ,{\omega }^{\ell },Z_1,\dots ,Z_k\right)\\ & +\sum _{i=1}^{k}F\left({\omega }^1,\dots ,{\omega }^{\ell },Z_1,\dots ,R(X,Y)Z_i,\dots ,Z_k\right)\end{array}$$(9)

例 1.9 (0 阶张量的曲率). $$R\left({\partial }_i,{\partial }_j\right)f={\nabla }_{{\partial }_i}\left({\nabla }_{{\partial }_j}f\right)-{\nabla }_{{\partial }_j}\left({\nabla }_{{\partial }_i}f\right)+{\nabla }_{\left[{\partial }_i,{\partial }_j\right]}f={\partial }_i{\partial }_{j}f-{\partial }_j{\partial }_{i}f=0$$

定义 1.10. 子丛的度量与联络 [BC2010].

定理 1.11 (度量的泰勒展式 [BC2010]). 在局部测地法坐标坐标下, $$g_{ij}\left(u^1,\dots ,u^n\right)={\delta }_{ij}-\frac{1}{3}{R}_{ikj\ell }{u}^k{u}^{\ell }+O\left(\Vert u{\Vert }^3\right).$$(10)

注 1.12. 法坐标下进行计算往往可以带来方便, 请参考 [Mei2013], P135-140.

2黎曼几何中的尺度分析

在黎曼流形 $(M^n,g)$ 上, 我们考虑对度量的一个尺度变换 $\bar{g}=\lambda g,\lambda >0$ 为常数.

首先, 黎曼联络 $\bar{\nabla }=\nabla$, 那么 ${\bar{R}}_{xy}z=R_{xy}z$.

截面曲率: 由$$R(x,y,z,w)=⟨R_{xy}w,z{⟩}_g,$$(11)知$$\bar{R}(x,y,z,w)=\lambda R(x,y,z,w).$$(12)而截面曲率$$k=R(x,y,x,y)=⟨R_{xy}y,x{⟩}_g,$$(13)其中 $|x{|}_g=|y{|}_g=1,⟨x,y{⟩}_g=0$. 所以$$|\frac{x}{\sqrt{\lambda }}{|}_{\bar{g}}=|\frac{y}{\sqrt{\lambda }}{|}_{\bar{g}}=1,⟨\frac{x}{\sqrt{\lambda }},\frac{y}{\sqrt{\lambda }}{⟩}_{\bar{g}}=0.$$(14)于是$$\bar{k}=\bar{R}(\frac{x}{\sqrt{\lambda }},\frac{y}{\sqrt{\lambda }},\frac{x}{\sqrt{\lambda }},\frac{y}{\sqrt{\lambda }})=(\frac{1}{\sqrt{\lambda }}{)}^4⟨R_{xy}y,x{⟩}_{\bar{g}}=\frac{1}{\lambda }⟨R_{xy}y,x{⟩}_g=\frac{1}{\lambda }k.$$(15)

Ricci 曲率:$$\mathrm{Ric}(x,y)=\sum _{i}R(x,e_i,y,e_i),$$(16)其中 $\{e_i\}$ 为相对于 $g$ 的一组幺正基, $x,y$ 为任意的向量. 那么$$\overline{\mathrm{Ric}}(x,y)=\frac{1}{\lambda }\sum _i\bar{R}(x,e_i,y,e_i)=\frac{1}{\lambda }\sum _i⟨R_{x{e}_i}{e}_i,y{⟩}_{\bar{g}}=\mathrm{Ric}(x,y).$$(17)而 $\mathrm{Ric}\ge (n-1)c$ 意味着 $\mathrm{Ric}(x,x)\ge (n-1)c$, 对任意的 $|x{|}_g=1$ 成立. 所以此时$$\overline{\mathrm{Ric}}(\frac{x}{\sqrt{\lambda }},\frac{x}{\sqrt{\lambda }})=\mathrm{Ric}(\frac{x}{\sqrt{\lambda }},\frac{x}{\sqrt{\lambda }})\ge (n-1)\frac{c}{\lambda },$$(18)也即 $\overline{\mathrm{Ric}}\ge (n-1)\frac{c}{\lambda }$.

数量曲率:$$\bar{S}=\frac{1}{{\lambda }^2}\sum _{ij}\bar{R}(e_i,e_j.e_i,e_j)=\frac{1}{\lambda }S.$$(19)

长度: 由$$d(p,q)={\int }_0^b|\dot{\gamma }{|}_{g}dt,$$(20)可知$$\bar{d}=\sqrt{\lambda }d.$$(21)

体积: 设相对于 $g$ 的体积形式为 $\Omega$, 由定义 $\Omega (e_1,...,e_n)=1$, 由此可见 $\overline{\Omega }={\sqrt{\lambda }}^n\Omega$, 所以体积$$\overline{V}={\sqrt{\lambda }}^{n}V.$$(22)

Laplace 算子与特征值: 因为$$\Delta f=g^{ij}(\frac{{\partial }^{2}f}{\partial x^i\partial x^j}-{\Gamma }_{ij}^k\frac{\partial f}{\partial x^k}),$$(23)而 ${\overline{\Gamma }}_{ij}^k={\Gamma }_{ij}^k$, 所以$$\overline{\Delta }f=\frac{1}{\lambda }\Delta f.$$(24)如果 $\mu$ 为 $\Delta$ 的特征值, 那么对应的 $\overline{\Delta }$ 的特征值为$$\bar{\mu }=\frac{\mu }{\lambda }.$$(25)

例 2.1. 正规化的 Sobolev 不等式$${\int }_{B_p(\rho )}|\nabla \varphi {|}^2\ge \frac{C_{SD}}{{\rho }^2}{\left({\int }_{B_p(\rho )}{\varphi }^{2\mu }\right)}^{\frac{1}{\mu }}$$以及$${\left({\int }_{B_p(\rho )}|f{|}^{2\mu /(\mu -2)}\right)}^{(\mu -2)/\mu }\le C_1{V}_p(\rho {)}^{-2/\mu }{\rho }^2\times {\int }_{B_p(\rho )}|\nabla f{|}^2+C_2{V}_p(\rho {)}^{-2/\mu }{\int }_{B_p(\rho )}{f}^2$$都是尺度不变的.

3微分形式

定义 3.1 (外积与反对称化算子). $$\text{ Alt }\alpha =\frac{1}{k!}\sum _{\sigma \in S_k}(\mathrm{sgn}\sigma )\left({}^{\sigma }\alpha \right),$$$$\omega \wedge \eta =\frac{(k+l)!}{k!l!}\mathrm{Alt}(\omega \otimes \eta ).$$

命题 3.2. $$\Phi \wedge \Psi \wedge \eta =\frac{(r+s+t)!}{r!s!t!}\mathrm{Alt}(\Phi \otimes \Psi \otimes \eta ).$$(26)

命题 3.3. $${\omega }^{i_1}\wedge {\omega }^{i_2}\wedge \cdots \wedge {\omega }^{i_r}=r!\mathrm{Alt}\left({\omega }^{i_1}\otimes {\omega }^{i_2}\otimes \cdots \otimes {\omega }^{i_r}\right).$$(27)

下面罗列一些微分形式的性质.

命题 3.4 ([Lee2012]). 设 $V$ 为 $n$ 维线性空间.

1.	$\omega \wedge \eta =(-1{)}^{kl}\eta \wedge \omega$.
2.	${\omega }^1\wedge \cdots \wedge {\omega }^k\left(v_1,\dots ,v_k\right)=\det \left({\omega }^j\left(v_i\right)\right)$.
3.	$\dim {\Lambda }^k\left(V^{\ast }\right)=\left(\begin{array}{l}n\\ k\end{array}\right)=\frac{n!}{k!(n-k)!}$.
4.	$\omega \left(T{v}_1,\dots ,T{v}_n\right)=(\det T)\omega \left(v_1,\dots ,v_n\right)$.

例 3.5. 记 $\omega ={\sum }_I^{\prime }{\omega }_{I}d{x}^{i_1}\wedge \cdots \wedge d{x}^{i_k}={\sum }_I^{\prime }{\omega }_{I}d{x}^I$, 那么由$$d{x}^{i_1}\wedge \cdots \wedge d{x}^{i_k}\left(\frac{\partial }{\partial x^{j_1}},\dots ,\frac{\partial }{\partial x^{j_k}}\right)={\delta }_J^I$$知$${\omega }_I=\omega \left(\frac{\partial }{\partial x^{i_1}},\dots ,\frac{\partial }{\partial x^{i_k}}\right).$$

定义 3.6 (内插 (内积)). 定义 $i_v:{\Lambda }^k\left(V^{\ast }\right)\to {\Lambda }^{k-1}\left(V^{\ast }\right)$, $v┘\omega =i_v\omega$ 为$$i_v\omega \left(w_1,\dots ,w_{k-1}\right)=\omega \left(v,w_1,\dots ,w_{k-1}\right).$$

命题 3.7. 设 $\omega \in {\Lambda }^k\left(V^{\ast }\right)$ 以及 $\eta \in {\Lambda }^l\left(V^{\ast }\right)$, 那么

1.	$i_v\circ i_v=0$.
2.	$i_v(\omega \wedge \eta )=\left(i_v\omega \right)\wedge \eta +(-1{)}^k\omega \wedge \left(i_v\eta \right)$.

命题 3.8 (拉回公式). 设 $F:M\to N$. 那么

1.	$F^{\ast }(\omega \wedge \eta )=\left(F^{\ast }\omega \right)\wedge \left(F^{\ast }\eta \right)$.
2.	$F^{\ast }\left({\sum }_I^{\prime }{\omega }_{I}d{y}^{i_1}\wedge \cdots \wedge d{y}^{i_k}\right)={\sum }_I^{\prime }\left({\omega }_I\circ F\right)d\left(y^{i_1}\circ F\right)\wedge \cdots \wedge d\left(y^{i_k}\circ F\right).$

命题 3.9 (最高阶形式的拉回). 设 $(x^i),(y^i)$ 分别为 $M,N$ 的局部坐标. 那么$$F^{\ast }\left(ud{y}^1\wedge \cdots \wedge d{y}^n\right)=(u\circ F)(\det DF)d{x}^1\wedge \cdots \wedge d{x}^n.$$

推论 3.10. $$d{\tilde{x}}^1\wedge \cdots \wedge d{\tilde{x}}^n=\det \left(\frac{\partial {\tilde{x}}^j}{\partial x^i}\right)d{x}^1\wedge \cdots \wedge d{x}^n.$$

定义 3.11 (外微分定义). $$d\left(\sum _J^{\prime }{\omega }_{J}d{x}^{j_1}\wedge \cdots \wedge d{x}^{j_k}\right)=\sum _J^{\prime }\sum _i\frac{\partial {\omega }_J}{\partial x^i}d{x}^i\wedge d{x}^{j_1}\wedge \cdots \wedge d{x}^{j_k}.$$

命题 3.12. 设 $\omega \in {\Lambda }^k\left(V^{\ast }\right)$ 以及 $\eta \in {\Lambda }^l\left(V^{\ast }\right)$, 那么

1.	$d(\omega \wedge \eta )=d\omega \wedge \eta +(-1{)}^k\omega \wedge d\eta$.
2.	$d\circ d\equiv 0$.
3.	$F^{\ast }(d\omega )=d\left(F^{\ast }\omega \right)$.

命题 3.13. 设 $\omega$ 为奇数阶的微分形式, 那么 $\omega \wedge \omega =0$.

命题 3.14 (微分形式的局部表示). 在局部坐标下, 微分形式可以表示为$$\begin{array}{rl}\alpha & =\sum _{i<}{\alpha }_{i_1\cdots i_r}d{x}^{i_1}\wedge \cdots \wedge d{x}^{i_r}\\ & =\frac{1}{r!}{\alpha }_{i_1\cdots i_r}d{x}^{i_1}\wedge \cdots \wedge d{x}^{i_r}\\ & ={\alpha }_{i_1\cdots i_r}d{x}^{i_1}\otimes \cdots \otimes d{x}^{i_r}.\end{array}$$(28)

定义 3.15 (微分形式的内积). $r$ 阶微分形式 $\alpha ,\beta$ 的内积定义为$$\begin{array}{r}⟨\alpha ,\beta ⟩\stackrel{\text{ def }}{=}\sum _{i<}{\alpha }^{i_1\cdots i_r}{\beta }_{i_1\cdots i_r}\\ =\frac{1}{r!}{\alpha }^{i_1\cdots i_r}{\beta }_{i_2\cdots i_r}.\end{array}$$(29)

定义 3.16 ($L^2$ 内积). $$(\phi ,\psi )={\int }_M⟨\phi ,\psi ⟩d{V}_M.$$

注 3.17. $$\mathrm{d}{x}^{i_1}\wedge \cdots \wedge \mathrm{d}{x}^{i_r}=\frac{1}{r!}{\delta }_{j_1\cdots j_r}^{i_1\cdots i_r}\mathrm{d}{x}^{j_1}\wedge \cdots \wedge \mathrm{d}{x}^{j_r}$$.

定义 3.18 (协变张量的散度). $$\mathrm{div}(\alpha {)}_{i_1\cdots i_{p-1}}\doteqdot g^{jk}{\nabla }_j{\alpha }_{k{i}_1\cdots i_{p-1}}={\nabla }_j{\alpha }_{j{i}_1\cdots i_{p-1}}$$(30)

例 3.19. 设 $\alpha$ 为 1 形式, 则 $\delta (\alpha )=-g^{ij}{\nabla }_i{\alpha }_j.$

命题 3.20. $${\nabla }_X(\theta \wedge \omega )={\nabla }_X\theta \wedge \omega +\theta \wedge {\nabla }_X\omega$$(31)

命题 3.21 (外微分的不变公式). $$\begin{array}{rl}d\omega & \left(X_1,\dots ,X_{k+1}\right)\\ & =\sum _{1\le i\le k+1}(-1{)}^{i-1}{X}_i\left(\omega \left(X_1,\dots ,\widehat{X_i},\dots ,X_{k+1}\right)\right)\\ & +\sum _{1\le i<j\le k+1}(-1{)}^{i+j}\omega \left(\left[X_i,X_j\right],X_1,\dots ,\widehat{X_i},\dots ,\widehat{X_j},\dots ,X_{k+1}\right).\end{array}$$(32)特别, 对于 $1$-形式 $\omega$: $$d\omega (X,Y)=X(\omega (Y))-Y(\omega (X))-\omega ([X,Y])$$(33)

命题 3.22 (外微分与联络). 利用联络的无挠性可知: $$d\theta \left(X_1,\cdots ,X_{r+1}\right)=\sum _{\alpha =1}^{r+1}(-1{)}^{\alpha +1}\cdot {\nabla }_{X_{\alpha }}\theta \left(X_1,\cdots ,{\hat{X}}_{\alpha },\cdots ,X_{r+1}\right).$$(34)

命题 3.23. $$(d\beta {)}_{i_0{i}_1\cdots i_p}=\sum _{j=0}^p(-1{)}^j{\nabla }_{i_j}{\beta }_{i_0{i}_1\cdots {\hat{i}}_j\cdots i_p}.$$(35)

例 3.24. 对于 1 形式和 2 形式分别有$$(d\beta {)}_{ij}=\left({\nabla }_i{\beta }_j-{\nabla }_j{\beta }_i\right),$$(36)$$(d\beta {)}_{ijk}=\left({\nabla }_i{\beta }_{jk}+{\nabla }_j{\beta }_{ki}+{\nabla }_k{\beta }_{ij}\right)$$(37)

注 3.25. 更多关于微分形式的 $L^2$ 内积的公式计算可以参考 [Jost2017] 的 140-103 页.

参考文献

[EL83]	James Eells and Luc Lemaire. Selected Topics in Harmonic Maps, 1983.
[BC2010]	Ben Andrews and Christopher Hopper. The Ricci flow in Riemannian geometry: a complete proof of the differentiable 1/4-pinching sphere theorem. springer, 2010.
[Mei2013]	梅加强. 流形与几何初步. 科学出版社, 2013.
[Lee2012]	John M Lee. Introduction to Smooth manifolds, volume 218. Springer, 2012.
[Jost2017]	Jürgen Jost. Riemannian geometry and geometric analysis. Universitext. Cham: Springer, 7th edition, 2017.