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\newcommand{\pt}{\left.\frac{\partial}{\partial\theta}\right|_{\gamma(t)}}

\newcommand{\dt}{\frac{\mathrm d}{\mathrm{d} t}}

\newcommand{\prr}{\frac{\partial}{\partial r}}

\newcommand{\ptt}{\frac{\partial}{\partial \theta}}

$

摘要 . In this paper, we solve the geodesics equation in the geodesic polar coordinates of a two dimensional Riemannian manifolds of constant sectional curvature. The relation between edges and angles of geodesic triangle has obtained and as a result the trigonometric formulae has been derived, that is the law of sines, the law of cosines.

1. Induced Connection Along a Mapping Suppose $M$ and $N$ be two smooth manifolds, and $\phi\mathpunct{:}N\to M$ is a smooth mapping. A vector field $X$ along $\phi$ is an assignment which corresponding each $x\in N$ to a vector $X(x)\in T_{\phi(x)}M$. In particular, for any vector field $V$ on $N$, $\phi_\ast V$ may not be a vector field on $M$, but it is a vector field along $\phi$. Clearly, the collection of vector fields along $\phi$ is a vector space, with the natural defined addtion and scalar multiplication.

We can extend a connection of vector fields on $M$ to a connection of vector fields along $\phi$. Let $\nabla$ be a connection of $M$, and $\set{E_i}_{i=1}^{\mathrm{dim} M}$ be a local frame in a neighborhood $\mathcal U$ of $\phi(x)$. Then the vector field $X$ along $\phi$ can be expressed as

\[

X(x)=X^i(x)E_i(\phi(x)),

\]

and we call $X$ is smooth if all the component functions $X^i$ are smooth. Now, taken a $v\in T_xN$, we define a mapping, which is called a covariant derivative, $\tilde\nabla$ from the vector fields along $\phi$ to $T_{\phi(x)}M$ as:

\[

\tilde\nabla_{v}X=(vX^i)E_i(\phi(x))+X^i(x)\nabla_{\phi_\ast v}E_i,

\]

a directly verification will show that it is independent on the choice of the frame. Let $V$ be a smooth vector field of $N$, then we obtain a smooth vector field along $\phi$ as

\[

(\tilde\nabla_VX)(x)=\tilde\nabla_{V(x)}X.

\]

The mapping $\tilde\nabla$ which assign a vector field $V$ of $N$ and a vector field $X$ along $\phi$ to the vector field along $\phi$, i.e., $\tilde\nabla_V X$, is called induced connection.

Particularly, if $\gamma$ is a curve on $M$, then the induced connection is commonly referred to as the connection along a curve [1].

An important property of the induce connection is that, even though we will not use it, if $\nabla$ is the Riemannian connection then $\tilde\nabla$ will also be “Riemannian” in the following sense\footnote{the second one is a litter trick, for a proof, see [1]}:

\begin{align*}

v\langle X,Y\rangle&=\langle\tilde\nabla_vX,Y\rangle+\langle X,\tilde\nabla_vY\rangle,\\

\phi_\ast([V,W])&=\tilde\nabla_V\phi_\ast W+\tilde\nabla_W\phi_\ast V,

\end{align*}

where $v\in T_xN$, $X,Y$ are vector fields along $\phi$ and $V,W$ are vector fields on $N$.

2. The Geodesics in Geodesic Polar Coordinates
At once one have a connection along a curve, then we can define the concept of parallel transformation along the curve, and obtain the geodesics of a manifold as the self-parallel curve. More precisely, if $\gamma\mathpunct{:}[0,1]\to M$ is a curve, and the tangent vector field of $\gamma(t)$ will be denoted as $\dot\gamma(t)=\gamma_\ast\left(\dt\right)$, which is clearly a vector field along $\gamma$, then

\[

\nabla_{\dot\gamma}\dot\gamma=0.

\]

The above equation is a second-order non-linear ordinary differential equation (in a local frame), by the theory of ordinary differential equations, it will have an unique solution for the following initial values

\[

\begin{cases}

\gamma(0)=p\\

\dot\gamma(0)=v,

\end{cases}

\]

where $p\in M, v\in T_pM$. What is more, the fundamental theory of ordinary differential equation tells us that, for any $\eps>0$, by rescalling the time parameter $t$ then we can suppose that $\eps>1$, there exists a neighborhood $\mathcal U$ of $p$ and a $\delta>0$, such that for any $q\in\mathcal U$ and any $v\in T_qM$ with $|v|<\delta$, the geodesic equation will have a unique solution on $(-\eps,\eps)$ which satisfy the following initial condition
\[
\begin{cases}
\gamma(0)=q\\
\dot\gamma(0)=v.
\end{cases}
\]
This derive a map $\exp_p$ form a neighborhood of $T_pM$ to $M$, and in fact it is a locally diffeomorphism, called exponential map. Hence we can obtain a coordinate system in a neighborhood of $M$, which is induced from the coordinates of $T_pM$ by the exponential map. Especially, when we taken the polar coordinate systems on $T_pM$ then the corresponding coordinate will be called geodesic polar coordinates.
From now on, let $M$ be a two dimensional manifold of constant curvature, $\nabla$ be the Riemannian connection (Levi-Civita connection). The polar coordinates of $M$ at $p$ will be donoted as $(r,\theta)$, as usual, $\set{\frac{\partial}{\partial r},\frac{\partial}{\partial \theta}}$ denote the local natural basis. Then a curve $\gamma(t)$ on $M$ can be written as $(r(t),\theta(t))$, where $t$ is the arc-lenth parameter, and the tangent vector of $\gamma$ is $\dot\gamma(t)=\dot r(t)\pr+\dot\theta(t)\pt$. Now, if $\gamma(t)$ is a geodesic, then
\begin{align}
0&=\nabla_{\dot\gamma(t)}\dot\gamma(t)
=\nabla_{\dot\gamma(t)}\left(\dot r(t)\pr\right)+
\nabla_{\dot\gamma(t)}\left(\dot\theta(t)\pt\right)\notag\\
&=\dt\dot r(t)\cdot\pr+\dot r(t)\cdot\nabla_{\dot\gamma(t)}\prr
+\dt\dot\theta(t)\cdot\pt+\dot\theta(t)\cdot\nabla_{\dot\gamma(t)}\ptt\notag\\
&=\ddot r(t)\pr+\ddot\theta(t)\pt+\dot r(t)\left(\dot r(t)\nabla_{\pr}\prr+\dot\theta(t)\nabla_{\pt}\prr\right)\notag\\
&\qquad+\dot\theta(t)\left(\dot r(t)\nabla_{\pr}\ptt+\dot\theta(t) \nabla_{\pt}\ptt\right)\notag\\
&=\ddot r(t)\pr+\ddot\theta(t)\pt+2\dot r(t)\dot\theta(t)\nabla_{\pt}\prr+\dot\theta(t)\dot\theta(t) \nabla_{\pt}\ptt\label{eq:1}.
\end{align}
The last equation obtained, since the $r$-curves are geodesics, thus $\nabla_{\prr}\prr=0$ and note also that $[\prr,\ptt]=0$, hence $\nabla_{\ptt}\prr=\nabla_{\prr}\ptt$.
the Gauss lemms assert that $\langle\prr,\ptt\rangle=0$, then the metric of $M$ can be written as
\begin{equation}\label{eq:2}
(\mathrm ds)^2=(\mathrm d r)^2+f^2(r,\theta)(\mathrm d\theta)^2,
\end{equation}
where $f(r,\theta)$ is a positive function. Using the compatibility and torsion free again, we have
\begin{alignat*}{2}
\langle\nabla_{\ptt}\prr,\prr\rangle&=0,\quad&\langle\nabla_{\ptt}\prr,\ptt\rangle&=\langle\nabla_{\prr}\ptt,\ptt\rangle=ff_r,\\
\langle\nabla_{\ptt}\ptt,\prr\rangle&=-ff_r,\quad&
\langle\nabla_{\ptt}\ptt,\ptt\rangle&=ff_\theta.
\end{alignat*}
Inserting these relations into \eqref{eq:1},
\[
0=\left\{\ddot r(t)-\dot\theta(t)\dot\theta(t)ff_r\right\}\pr+
\left\{\ddot\theta(t)+2\dot r(t)\dot\theta(t)ff_r+\dot\theta(t)\dot\theta(t)ff_\theta\right\}\pt.
\]
Thus, the geodesic equation is
\begin{equation}\label{eq:3}\begin{cases}
0&=\ddot r(t)-\dot\theta(t)\dot\theta(t)ff_r\\
0&=\ddot\theta(t)+2\dot r(t)\dot\theta(t)ff_r+\dot\theta(t)\dot\theta(t)ff_\theta,
\end{cases}\end{equation}
where the $f,f_r$ should be evaulated at $\gamma(t)=(r(t),\theta(t))$.
To get the relation of $f$ should be satisfied as $M$ is of constant curvature, we can calculate the Christoffel symbols, but a more efficient way is the method moving frame.
Set $\omega^1=\rd r$, $\omega^2=f\rd\theta$, then \eqref{eq:2} can be re-written as
\[
(\rd s)^2=\omega^1\omega^1+\omega^2\omega^2.
\]
A directly calculation will show that
\[\begin{cases}
\rd\omega^1&=\rd(\rd r)=0\\
\rd\omega^2&=\rd(f\rd\theta)=f_r\rd r\wedge\rd\theta,
\end{cases}\]
thus, if we set $\omega_1^2=f_r\rd\theta=-\omega_2^1$, then
\[\begin{cases}
\rd\omega^1&=\omega^2\wedge\omega_2^1\\
\rd\omega^2&=\omega^1\wedge\omega_1^2，
\end{cases}\]
thus the Cartan's Lemma asserts that $\omega^2_1$ is the connection 1-form and the Gauss equation says that
\[
\rd\omega_1^2=-K\omega^1\wedge\omega^2,
\]
where $K$ is the Gauss curvature of $M$, which is a constant by assumption.
We conclude that for a 2-dimensional manifold $M$, it has constant sectional curvature if and only if the function $f$ must satisfy the following differential equation
\begin{equation}\label{eq:4}
f_{rr}+K f=0.
\end{equation}
Hence, since $t$ is arc-length parameter, the geodesic equations are
\begin{equation}
\begin{cases}
0&=\ddot r(t)-\dot\theta(t)\dot\theta(t)ff_r\\
0&=\ddot\theta(t)+2\dot r(t)\dot\theta(t)ff_r+\dot\theta(t)\dot\theta(t)ff_\theta\\
1&=\dot r(t)^2+f^2\dot\theta(t)^2\\
0&=f_{rr}+K f.
\end{cases}
\end{equation}

参考文献

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