Machine-translated from Chinese.
线性联络
定义
回忆$C^\infty$流形$M$上黎曼度量$g$指对每个$T_xM$指定一个向量内积$g_x(-,-),$
也记为$\left<{}-,-\right>_x.$ 称它是$C^\infty$的,
若任取坐标系$(x^1,\cdots,x^n),$
$g_{ij}(x)=g_{x}(\frac{\partial {} }{\partial {}x^i},\frac{\partial {} }{\partial {}x^j})$都是$C^\infty$函数,
$1\le i,j\le n.$
在$\mathbb{R}^n$中,
对向量场$X=\sum_i X^i\frac{\partial {} }{\partial {}x^i},$
沿$v$的方向导数$D_vX:=\sum_i D_vX^i\frac{\partial {} }{\partial {}x^i}.$
若$V$是一个向量场, 那么可以定义新的向量场$D_VX(x)=D_{V(x)}X,$
称为方向导数, 满足如下性质:
$D_{fV+gW}X=fD_VX+gD_WX;$
$D_V(fX)=(D_Vf)X+fD_VX;$
$D_V(X+Y)=D_VX+D_VY.$
定义 1. 在$C^\infty$流形$M$上, 联络即对每一对$C^\infty$向量场$V,X,$ 指定新的$C^\infty$向量场$D_VX,$ 满足如上三条性质. $D_Vf$视为$Vf.$
$\mathbb{R}^n$中的方向导数为标准联络.
联络联系了$M$中各点的切空间. 称$D_VX$为$X$沿$V$的协变导数,
$D$有时称为协变微分.
由于第一条性质说明$D_VX$对$V$是$C^\infty$函数环$\mathscr{F}(M)$上线性的,
有时强调此时定义的联络为线性联络, 区别于仿射联络, 射影联络.
线性联络满足如下性质. 若$x\in M$有$V(x)=W(x),$ 那么$D_VX(x)=D_WX(x).$
由此可定义$D_vX:=D_VX(x)\in T_xM,$ $V$为任意满足$V(x)=v$的向量场.
但是对$X(x)=Y(x),$ $D_VX(x)$并不见得等于$D_VY(x).$ 不过若$V(x)=v,$
在从$x$出发以$v$为切方向的一小段弧线$\gamma$上有$X|_{\gamma}=Y|_{\gamma},$
那么可以证明$D_VX(x)=D_VY(x).$ 因此可以说$D_vX$由$X|_{\gamma}$决定.
通过将标准联络拉回, 局部总是可以定义线性联络. 由单位分解,
整体联络也是可以给出的, 因为流形上联络全体组成凸集. 由此可以看出,
联络有很多. 我们希望找到有意思的联络.
Levi-Civita联络
命题 2. 对$M$上给定的黎曼度量$g,$ 存在$M$上的唯一联络$D,$ 使得对任意向量场$X,Y,Z,$ 满足如下两条性质:
$X\left<{}Y,Z\right>=\left<{}D_XY,Z\right>+\left<{}Y,D_XZ\right>;$
$D_XY-D_YX=[X,Y]=XY-YX.$
这样的联络被称为度量$g$的Levi-Civita联络或度量$g$的黎曼联络.
它与后面说明的”平行移动”的性质等价.
第一条性质给出了对内积求导的方法, 而第二条性质满足与Lie括号定义匹配.
证: 首先证明唯一性. 由如下方式, 给出函数$\Gamma^k_{ij}\in C^\infty:$
$$
D_{\frac{\partial {} }{\partial {}x^i} }\frac{\partial {} }{\partial {}x^j}=\sum_{k=1}^n \Gamma_{ij}^k \frac{\partial {} }{\partial {}x^k},\quad 1\le i,j\le n.
$$
我们只需说明联络系数$\Gamma_{ij}^k$由度量$g$唯一确定.
由性质(L1), 以后默认采用Einstein求和约定, 我们有
$$
\frac{\partial {}g_{jk} }{\partial {}x^i}=\frac{\partial {} }{\partial {}x^i}\left<{}\frac{\partial {} }{\partial {}x^j},\frac{\partial {} }{\partial {}x^k}\right>=
\left<{}\Gamma_{ij}^l\frac{\partial {} }{\partial {}x^l},\frac{\partial {} }{\partial {}x^k}\right>+\left<{}\frac{\partial {} }{\partial {}x^j},\Gamma_{ik}^l\frac{\partial {} }{\partial {}x^l}\right>=\Gamma_{ij}^l g_{lk}+\Gamma_{ik}^l g_{jl} \tag*{(*)}
$$
由性质(L2),
$$
(\Gamma_{ij}^k-\Gamma_{ji}^k)\frac{\partial {} }{\partial {}x^k}=\left[\frac{\partial {} }{\partial {}x^i},\frac{\partial {} }{\partial {}x^j}\right]=0 \,\Rightarrow\, \Gamma_{ij}^k=\Gamma_{ji}^k.
$$
轮换$(\ast )$式$i,j,k$, 得到
$$
\frac{\partial {}g_{jk} }{\partial {}x^i}+\frac{\partial {}g_{ik} }{\partial {}x^j}-\frac{\partial {}g_{ij} }{\partial {}x^k}=2\Gamma_{ij}^lg_{lk}\,\Rightarrow\, \Gamma_{ij}^k=\frac{g^{kl} }{2}\left(\frac{\partial {}g_{jl} }{\partial {}x^i}+\frac{\partial {}g_{il} }{\partial {}x^j}-\frac{\partial {}g_{ij} }{\partial {}x^l}\right).
$$
其中$(g^{ij})$为$(g_{ij})$的逆阵, 即满足$g^{ik}g_{kj}=\delta^i_j,$
$\,\forall\,i,j.$
上式说明$\Gamma_{ij}^k$由度量$g$决定, 这说明了唯一性. 至于存在性,
只需说明由$\Gamma_{ij}^k$反过来也可以给出一个联络.
对向量场$X=f^i\frac{\partial {} }{\partial {}x_i},$
$Y=h^i\frac{\partial {} }{\partial {}x_i},$ 由联络性质, 自然定义
$$
D_XY=f^i\frac{\partial {}h^j}{\partial {}x_i}\frac{\partial {} }{\partial {}x_j}+f^ih^j \left(D_{\frac{\partial {} }{\partial {}x_i} }\frac{\partial {} }{\partial {}x_j}\right)=f^i\left(\frac{\partial {}h^j}{\partial {}x_j}+h^k \Gamma_{ik}^j\right)\frac{\partial {} }{\partial {}x_j}.
$$
验证发现其与坐标系选取无关, 因此向量场良定. 由定义过程,
它当然满足联络的性质, 从而$D$即为所求, 存在性得证.
证明中用度量来表示联络系数的方式由 Christoffel 得到,
因此将 Levi-Civita 联络的$\Gamma_{ij}^k$称为 Christoffel 记号.
不借助坐标系, 也可通过Levi-Civita联络的性质得到如下表示:
$$
\left<{}D_XY,Z\right>=\frac{1}{2}\left(X\left<{}Y,Z\right>+Y\left<{}Z,X\right>-Z\left<{}X,Y\right>+\left<{}Z,[X,Y]\right>+\left<{}Y,[Z,X]\right>-\left<{}X,[Y,Z]\right>\right).
$$
这完全确定了Levi-Civita联络. 当$X,Y,Z$满足好的性质时, 如彼此正交,
则上式的表述将大幅简化.
几何意义
我们知道, 子流形$N$上可以继承外围空间$M$中原有的黎曼度量$g$,
得到诱导度量$\overline{g}.$ 一个好的事实是, $\,\forall\,p\in N,$
令$P:T_pM\rightarrow T_pN$为正交投影,
则$\,\forall\,X,Y\in \mathfrak{X}(N)\subset \mathfrak{X}(M),$
定义$\overline{D}_XY=P(D_XY),$
则$\overline{D}$恰为$N$上度量$\overline{g}$的Levi-Civita联络.
特别地, 当外围空间$M=\mathbb{R}^n$时, 其有自然的平坦度量$g,$
自然基标准正交. 易验证一开始提到的方向导数就是它的Levi-Civita联络.
因此上述事实表明了$N\subset \mathbb{R}^n$时Levi-Civita联络的几何意义:
它是外围空间中向量场方向导数的投影.
可以看出, 性质(L2)在没有给出黎曼度量的情况下也可以讨论.
它等价于$\Gamma_{ij}^k=\Gamma_{ji}^k,$ $\,\forall\,1\le i,j,k\le n.$
因此称满足(L2)的联络是对称的, 也称为无挠的.
我们还需更直观的理解联络. 设$\gamma:[a,b]\rightarrow M$为一条嵌入曲线,
像落在某坐标邻域中. 设$X$为$\gamma(t)$(邻域)上的向量场.
我们关心何时有$D_{\dot\gamma(t)}X=0,$ $\,\forall\,t\in [a,b].$
$$
\begin{aligned}
D_{\dot\gamma(t)}X&=D_{\dot\gamma(t)}\left(X^j\frac{\partial {} }{\partial {}x^j}\right)\\
&=\frac{d {}X^j}{d {}t}\frac{\partial {} }{\partial {}x_j}+\dot\gamma^iX^jD_{\frac{\partial {} }{\partial {}x_i} }\frac{\partial {} }{\partial {}x^j}\\
&=\left(\frac{d {}X^k}{d {}t}+\dot\gamma^iX^j\Gamma_{ij}^k\right)\frac{\partial {} }{\partial {}x^k}
\end{aligned}
$$
因此$D_{\dot\gamma(t)}X=0$即要求$\,\forall\,k,$
有$\frac{d {}X^k}{d {}t}+\dot\gamma^iX^j\Gamma_{ij}^k=0$成立.
这是一个ODE方程组, $\,\forall\,v\in T_{\gamma(a)}M,$ 总存在$X_v(t),$
满足$X_v(a)=v,$ $D_{\dot\gamma(t)}X_v=0.$
称一个沿$\gamma$的向量场$X$是平行的, 若$D_{\dot\gamma}X=0.$
给定$\gamma:[a,b]\rightarrow M,$
称$w\in T_{\gamma(b)}M$由$v\in T_{\gamma(a)}M$沿$\gamma$平行移动得到,
若存在沿$\gamma$的平行向量场$X,$ 使得$X(a)=v,$ $X(b)=w.$
以上都是对一小段嵌入曲线讨论的结果. 对一般的浸入曲线, 需要对区间做分割,
使向量分段平移.
由此我们可以定义向量空间的同构$P^\gamma:M_{\gamma(a)}\rightarrow M_{\gamma(b)},$
它的逆恰恰是沿$\gamma$反向曲线的平行移动. 称该同构为平移同构.
注意平移同构与曲线的选取密切相关. 因此,
我们可以说联络联系着曲线两端的切空间, 当然这一联系与曲线本身有关.
现在我们能对(L1)性质有更深刻的认识: 它等价于所有平移同构都保持内积.
当(L1)成立时, 设有弧长参数化曲线$\gamma:[0,1]\rightarrow M,$
$x,y$为其端点. 取$T_xM$标准正交基$\{e_i\},$
令$e_i(t)=P^{\gamma|_{[0,t]} }(e_i),$ 有$D_{\dot\gamma(t)}e_i(t)=0.$
因此,
$$
\frac{d {} }{d {}t}\left<{}e_i(t),e_j(t)\right>=\dot\gamma(t)\left<{}e_i(t),e_j(t)\right>=\left<{}D_{\dot\gamma(t)}e_i(t),e_j(t)\right>+\left<{}e_i(t),D_{\dot\gamma(t)}e_j(t)\right>=0.
$$
从而, $\left<{}e_i(t),e_j(t)\right>\equiv \delta_{ij}.$ 因此,
平移同构将标准正交基映为标准正交基, 从而是保持内积的同构. 反过来,
设每一个平移同构都保持内积, 取向量场$X,Y,Z.$ 对任意点$x,$
取弧长参数化曲线$\gamma:[0,1]\rightarrow M,$
以$x$为起点且$\dot\gamma(0)=X(x).$
类似地取$\{e_i(t)\}$为$T_{\gamma(t)}M$的标准正交基, 那么设$Y=Y^ie_i,$
$Z=Z^ie_i,$ 则:
$$
\begin{aligned}
X(x)\left<{}Y,Z\right>&=\dot\gamma(0)\left<{}Y,Z\right>=\frac{d {} }{d {}t}\sum_i Y^iZ^i|_{t=0}\\
&=\sum_i \frac{d {}Y^i}{d {}t}(0)Z^i(0) + \sum_i Y^i(0)\frac{d {}Z^i}{d {}t}(0)\\
&=\left<{}D_{\dot{\gamma}(0)}Y,Z\right>+\left<{}Y,D_{\dot{\gamma}(0)}Z\right>\\
&=\left<{}D_{X(x)}Y,Z\right>+\left<{}Y,D_{X(x)}Z\right>
\end{aligned}
$$
即若平移同构总保持内积, 那么它也一定满足(L1)条件. 于是,
可以说Levi-Civita联络是唯一的对称联络, 使得所有平移同构都是等距同构.
注 3. 联络$D$决定了平移同构, 反过来平移同构也决定了该联络: 可以证明, 若给定$v\in T_xM,$ 向量场$X,$ 曲线$\gamma$以$x$为起点, $\dot{\gamma}(0)=v,$ 那么对于平移同构$P_t:T_xM\rightarrow T_{\gamma(t)}M,$ 有
$$
D_vX=\frac{d {} }{d {}t}[P_t^{-1}(X\circ\gamma(t))]|_{t=0}.
$$
注 4. 测地线的概念由测地曲率为零引入, 即曲线的曲率向量与法线重合, 曲线在切平面上的投影处处曲率为零, 是直线的推广. 另一种推广方式是考虑曲线上切向量场, 如果切向量场是沿曲线的平行向量场, 那么曲线称为联络的测地线, 即$D_{\dot\gamma}\dot\gamma=0.$ 于是这两种定义是等价的, 且通过后一定义立即看出测地线的定义是内蕴的.
文章最后更新于 2022-01-14 12:22:00
Linear connection
definition
Recall that the Riemannian metric $g$ on $C^\infty$ manifold $M$ means specifying a vector inner product $g_x(-,-),$ for each $T_xM$
Also recorded as $\left<{}-,-\right>_x.$, it is called $C^\infty$,
If you choose any coordinate system $(x^1,\cdots,x^n),$
$g_{ij}(x)=g_{x}(\frac{\partial {} }{\partial {}x^i},\frac{\partial {} }{\partial {}x^j})$ are all $C^\infty$ functions,
$1\le i,j\le n.$
In $\mathbb{R}^n$,
For vector field $X=\sum_i X^i\frac{\partial {} }{\partial {}x^i},$
Directional derivative $D_vX:=\sum_i D_vX^i\frac{\partial {} }{\partial {}x^i}.$ along $v$
If $V$ is a vector field, then a new vector field $D_VX(x)=D_{V(x)}X,$ can be defined
called Directional derivative, satisfying the following properties:
$D_{fV+gW}X=fD_VX+gD_WX;$
$D_V(fX)=(D_Vf)X+fD_VX;$
$D_V(X+Y)=D_VX+D_VY.$
Definition 1. On $C^\infty$ manifold $M$, Contact That is, for each pair of $C^\infty$ vector fields $V,X,$, specify a new $C^\infty$ vector field $D_VX,$ that satisfies the above three properties. $D_Vf$ is regarded as $Vf.$
The directional derivative in $\mathbb{R}^n$ is Standard contact.
Contact Contact $M$ tangent space of every point in. Call $D_VX$ the edge of $X$ along $V$ covariant derivative,
$D$ is sometimes called a covariant differential.
Since the first property shows that $D_VX$ is linear to $V$ on the $C^\infty$ function ring $\mathscr{F}(M)$,
Sometimes it is emphasized that the contact defined at this time is Linear connection, which is different from affine connection and projective connection.
The linear connection satisfies the following properties. If $x\in M$ has $V(x)=W(x),$ then $D_VX(x)=D_WX(x).$
From this, $D_vX:=D_VX(x)\in T_xM,$ $V$ can be defined as any vector field that satisfies $V(x)=v$.
But $X(x)=Y(x),$ $D_VX(x)$ is not necessarily equal to $D_VY(x).$, but if $V(x)=v,$
On a short arc $\gamma$ starting from $x$ with $v$ as the tangent direction, there is $X|_{\gamma}=Y|_{\gamma},$
Then it can be proved that $D_VX(x)=D_VY(x).$ therefore can be said that $D_vX$ is determined by $X|_{\gamma}$.
By pulling back the standard connections, it is always possible to define linear connections locally. Decomposed by units,
The overall connection can also be given, because the entire connection on the manifold forms a convex set. It can be seen from this that,
There are many contacts. We hope to find interesting ones.
Levi-Civita Contact
Proposition 2. For a given Riemannian metric $g,$ on $M$, there is a unique connection $D,$ on $M$ such that the following two properties are satisfied for any vector field $X,Y,Z,$:
$X\left<{}Y,Z\right>=\left<{}D_XY,Z\right>+\left<{}Y,D_XZ\right>;$
$D_XY-D_YX=[X,Y]=XY-YX.$
Such a connection is called the metric $g$ Levi-Civita Contact or metric$g$ Riemann Contact.
It is equivalent to the property of "parallel movement" explained later.
The first property gives the method of deriving the inner product, while the second property matches the definition of Lie brackets.
Certificate: First prove the uniqueness. In the following way, the function $\Gamma^k_{ij}\in C^\infty:$ is given
$$
D_{\frac{\partial {} }{\partial {}x^i} }\frac{\partial {} }{\partial {}x^j}=\sum_{k=1}^n \Gamma_{ij}^k \frac{\partial {} }{\partial {}x^k},\quad 1\le i,j\le n.
$$
we just need to explain Contact coefficient $\Gamma_{ij}^k$ is uniquely determined by the metric $g$.
According to property (L1), the Einstein summation convention will be adopted by default in the future, we have
$$
\frac{\partial {}g_{jk} }{\partial {}x^i}=\frac{\partial {} }{\partial {}x^i}\left<{}\frac{\partial {} }{\partial {}x^j},\frac{\partial {} }{\partial {}x^k}\right>=
\left<{}\Gamma_{ij}^l\frac{\partial {} }{\partial {}x^l},\frac{\partial {} }{\partial {}x^k}\right>+\left<{}\frac{\partial {} }{\partial {}x^j},\Gamma_{ik}^l\frac{\partial {} }{\partial {}x^l}\right>=\Gamma_{ij}^l g_{lk}+\Gamma_{ik}^l g_{jl} \tag*{(*)}
$$
From the property (L2),
$$
(\Gamma_{ij}^k-\Gamma_{ji}^k)\frac{\partial {} }{\partial {}x^k}=\left[\frac{\partial {} }{\partial {}x^i},\frac{\partial {} }{\partial {}x^j}\right]=0 \,\Rightarrow\, \Gamma_{ij}^k=\Gamma_{ji}^k.
$$
Rotate $(\ast )$ formula $i,j,k$, get
$$
\frac{\partial {}g_{jk} }{\partial {}x^i}+\frac{\partial {}g_{ik} }{\partial {}x^j}-\frac{\partial {}g_{ij} }{\partial {}x^k}=2\Gamma_{ij}^lg_{lk}\,\Rightarrow\, \Gamma_{ij}^k=\frac{g^{kl} }{2}\left(\frac{\partial {}g_{jl} }{\partial {}x^i}+\frac{\partial {}g_{il} }{\partial {}x^j}-\frac{\partial {}g_{ij} }{\partial {}x^l}\right).
$$
Where $(g^{ij})$ is the inverse matrix of $(g_{ij})$, that is, $g^{ik}g_{kj}=\delta^i_j,$ is satisfied
$\,\forall\,i,j.$
The above formula shows that $\Gamma_{ij}^k$ is determined by the metric $g$, which illustrates the uniqueness. As for existence,
Suffice it to say that a connection can also be given by $\Gamma_{ij}^k$ in turn.
Pair vector field$X=f^i\frac{\partial {} }{\partial {}x_i},$
$Y=h^i\frac{\partial {} }{\partial {}x_i},$ Naturally defined by contact properties
$$
D_XY=f^i\frac{\partial {}h^j}{\partial {}x_i}\frac{\partial {} }{\partial {}x_j}+f^ih^j \left(D_{\frac{\partial {} }{\partial {}x_i} }\frac{\partial {} }{\partial {}x_j}\right)=f^i\left(\frac{\partial {}h^j}{\partial {}x_j}+h^k \Gamma_{ik}^j\right)\frac{\partial {} }{\partial {}x_j}.
$$
Verification shows that it has nothing to do with the selection of the coordinate system, so the vector field is well-defined. From the definition process,
It certainly satisfies the property of connection, so $D$ is what is desired, and its existence is proved.
The method of expressing the contact coefficient in terms of metrics in the proof was obtained by Christoffel,
Therefore, $\Gamma_{ij}^k$ of the Levi-Civita connection is called the Christoffel notation.
Without the help of a coordinate system, the following expression can also be obtained through the properties of Levi-Civita connection:
$$
\left<{}D_XY,Z\right>=\frac{1}{2}\left(X\left<{}Y,Z\right>+Y\left<{}Z,X\right>-Z\left<{}X,Y\right>+\left<{}Z,[X,Y]\right>+\left<{}Y,[Z,X]\right>-\left<{}X,[Y,Z]\right>\right).
$$
This completely determines the Levi-Civita connection. When $X,Y,Z$ satisfies good properties, such as being orthogonal to each other,
Then the expression of the above formula will be greatly simplified.
Geometric meaning
We know that the original Riemannian metric $g$ in the peripheral space $M$ can be inherited on the submanifold $N$,
A good fact to get the induced metric $\overline{g}.$ is, $\,\forall\,p\in N,$
Let $P:T_pM\rightarrow T_pN$ be Orthographic projection,
Then $\,\forall\,X,Y\in \mathfrak{X}(N)\subset \mathfrak{X}(M),$
Definition$\overline{D}_XY=P(D_XY),$
Then $\overline{D}$ is exactly the Levi-Civita connection measuring $\overline{g}$ on $N$.
In particular, when the peripheral space $M=\mathbb{R}^n$ is, it has a natural flat metric $g,$
Natural basis orthonormal. It is easy to verify that the directional derivative mentioned at the beginning is its Levi-Civita connection.
Therefore, the above facts indicate the geometric significance of the Levi-Civita connection at $N\subset \mathbb{R}^n$:
It is the projection of the directional derivative of the vector field in peripheral space.
It can be seen that property (L2) can also be discussed without giving the Riemannian metric.
It is equivalent to $\Gamma_{ij}^k=\Gamma_{ji}^k,$ $\,\forall\,1\le i,j,k\le n.$
Therefore, the connection satisfying (L2) is said to be Symmetry of, also known as Indestructible of.
We also need to understand the connection more intuitively. Let $\gamma:[a,b]\rightarrow M$ be an embedded curve,
The image falls in a certain coordinate neighborhood. Let $X$ be the vector field on $\gamma(t)$ (neighborhood).
We care about when $D_{\dot\gamma(t)}X=0,$ $\,\forall\,t\in [a,b].$
$$
\begin{aligned}
D_{\dot\gamma(t)}X&=D_{\dot\gamma(t)}\left(X^j\frac{\partial {} }{\partial {}x^j}\right)\\
&=\frac{d {}X^j}{d {}t}\frac{\partial {} }{\partial {}x_j}+\dot\gamma^iX^jD_{\frac{\partial {} }{\partial {}x_i} }\frac{\partial {} }{\partial {}x^j}\\
&=\left(\frac{d {}X^k}{d {}t}+\dot\gamma^iX^j\Gamma_{ij}^k\right)\frac{\partial {} }{\partial {}x^k}
\end{aligned}
$$
Therefore $D_{\dot\gamma(t)}X=0$ requires $\,\forall\,k,$
There is $\frac{d {}X^k}{d {}t}+\dot\gamma^iX^j\Gamma_{ij}^k=0$ established.
This is a system of ODE equations, $\,\forall\,v\in T_{\gamma(a)}M,$ always exists $X_v(t),$
Meet $X_v(a)=v,$ $D_{\dot\gamma(t)}X_v=0.$
A vector field $X$ along $\gamma$ is called parallel , if$D_{\dot\gamma}X=0.$
Given$\gamma:[a,b]\rightarrow M,$
It is said that $w\in T_{\gamma(b)}M$ is along $\gamma$ from $v\in T_{\gamma(a)}M$ parallel movement get,
If there is an edge along $\gamma$ parallel vector field $X,$ makes $X(a)=v,$ $X(b)=w.$
The above are the results of the discussion on a small section of embedded curve. For a general immersion curve, the interval needs to be divided,
Translate the vector piecewise.
From this we can define the isomorphism $P^\gamma:M_{\gamma(a)}\rightarrow M_{\gamma(b)},$ of the vector space
Its inverse is exactly the parallel movement along the reverse curve of $\gamma$. This isomorphism is called translational isomorphism.
Note that translation isomorphism is closely related to the selection of curves. Therefore,
we can say Contact the tangent space at both ends of the curve, of course this connection is related to the curve itself.
Now we can have a deeper understanding of the (L1) property: it is equivalent to preserving the inner product for all translational isomorphisms.
When (L1) is established, there is an arc length parameterized curve $\gamma:[0,1]\rightarrow M,$
$x,y$ is its endpoint. Take $T_xM$ orthonormal basis $\{e_i\},$
Let $e_i(t)=P^{\gamma|_{[0,t]} }(e_i),$ have $D_{\dot\gamma(t)}e_i(t)=0.$
Therefore,
$$
\frac{d {} }{d {}t}\left<{}e_i(t),e_j(t)\right>=\dot\gamma(t)\left<{}e_i(t),e_j(t)\right>=\left<{}D_{\dot\gamma(t)}e_i(t),e_j(t)\right>+\left<{}e_i(t),D_{\dot\gamma(t)}e_j(t)\right>=0.
$$
Thus, $\left<{}e_i(t),e_j(t)\right>\equiv \delta_{ij}.$ Therefore,
The translational isomorphism maps the orthonormal basis to the orthonormal basis, and is thus an isomorphism that preserves the inner product. In turn,
Assume that each translational isomorphism maintains the inner product, and takes the vector field $X,Y,Z.$ for any point $x,$
Get arc length parameterized curve $\gamma:[0,1]\rightarrow M,$
Take $x$ as the starting point and $\dot\gamma(0)=X(x).$
Similarly, take $\{e_i(t)\}$ as the orthonormal basis of $T_{\gamma(t)}M$, then let $Y=Y^ie_i,$
$Z=Z^ie_i,$ then:
$$
\begin{aligned}
X(x)\left<{}Y,Z\right>&=\dot\gamma(0)\left<{}Y,Z\right>=\frac{d {} }{d {}t}\sum_i Y^iZ^i|_{t=0}\\
&=\sum_i \frac{d {}Y^i}{d {}t}(0)Z^i(0) + \sum_i Y^i(0)\frac{d {}Z^i}{d {}t}(0)\\
&=\left<{}D_{\dot{\gamma}(0)}Y,Z\right>+\left<{}Y,D_{\dot{\gamma}(0)}Z\right>\\
&=\left<{}D_{X(x)}Y,Z\right>+\left<{}Y,D_{X(x)}Z\right>
\end{aligned}
$$
That is, if the translational isomorphism always maintains the inner product, then it must also satisfy the condition (L1). Therefore,
It can be said that the Levi-Civita connection is the only symmetric connection, making all translational isomorphisms isometric isomorphisms.
Note 3. The connection $D$ determines the translational isomorphism, and in turn the translational isomorphism also determines the connection: It can be proved that if $v\in T_xM,$ is given, the vector field $X,$ and the curve $\gamma$ take $x$ as the starting point, $\dot{\gamma}(0)=v,$ then for the translational isomorphism $P_t:T_xM\rightarrow T_{\gamma(t)}M,$ we have
$$
D_vX=\frac{d {} }{d {}t}[P_t^{-1}(X\circ\gamma(t))]|_{t=0}.
$$
Note 4. The concept of geodesic is introduced by the geodesic curvature being zero, that is, the curvature vector of the curve coincides with the normal, and the projection of the curve on the tangent plane has zero curvature everywhere, which is a generalization of the straight line. Another generalization method is to consider the tangent vector field on the curve. If the tangent vector field is a parallel vector field along the curve, then the curve is called a contact geodesic, that is, $D_{\dot\gamma}\dot\gamma=0.$ So these two definitions are equivalent, and through the latter definition it is immediately obvious that the definition of geodesic is intrinsic.
The article was last updated on 2022-01-14 12:22:00
