$\\newcommand{\\pr}{\\left.\\frac{\\partial}{\\partial r}\\right|_{\\gamma(t)}}
\n\\newcommand{\\pt}{\\left.\\frac{\\partial}{\\partial\\theta}\\right|_{\\gamma(t)}}
\n\\newcommand{\\dt}{\\frac{\\mathrm d}{\\mathrm{d} t}}
\n\\newcommand{\\prr}{\\frac{\\partial}{\\partial r}}
\n\\newcommand{\\ptt}{\\frac{\\partial}{\\partial \\theta}}
\n$
\n\\begin{abstract}
\n 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.
\n \\end{abstract}
\n \\section{Induced Connection Along a Mapping}
\n Suppose $M$ and $N$ be two smooth manifolds, and $\\phi\\mathpunct{:}N\\to M$ is a smooth mapping. A \\emph{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.
\n
\n 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
\n \\[
\n X(x)=X^i(x)E_i(\\phi(x)),
\n \\]
\n and we call $X$ is \\emph{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 \\emph{covariant derivative}, $\\tilde\\nabla$ from the vector fields along $\\phi$ to $T_{\\phi(x)}M$ as:
\n \\[
\n \\tilde\\nabla_{v}X=(vX^i)E_i(\\phi(x))+X^i(x)\\nabla_{\\phi_\\ast v}E_i,
\n \\]
\n 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
\n \\[
\n (\\tilde\\nabla_VX)(x)=\\tilde\\nabla_{V(x)}X.
\n \\]
\n 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 \\emph{induced connection}. <\/p>\n
Particularly, if $\\gamma$ is a curve on $M$, then the induced connection is commonly referred to as the connection along a curve \\cite{gallot2004riemannian}. <\/p>\n
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 \\cite{chen2004riemannian}}:
\n \\begin{align*}
\n v\\langle X,Y\\rangle&=\\langle\\tilde\\nabla_vX,Y\\rangle+\\langle X,\\tilde\\nabla_vY\\rangle,\\\\
\n \\phi_\\ast([V,W])&=\\tilde\\nabla_V\\phi_\\ast W+\\tilde\\nabla_W\\phi_\\ast V,
\n \\end{align*}
\n where $v\\in T_xN$, $X,Y$ are vector fields along $\\phi$ and $V,W$ are vector fields on $N$.<\/p>\n
\\section{The Geodesics in Geodesic Polar Coordinates}
\n At once one have a connection along a curve, then we can define the concept of \\emph{parallel transformation} along the curve, and obtain the geodesics of a manifold as the \\emph{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
\n \\[
\n \\nabla_{\\dot\\gamma}\\dot\\gamma=0.
\n \\]
\nThe 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
\n \\[
\n \\begin{cases}
\n \\gamma(0)=p\\\\
\n \\dot\\gamma(0)=v,
\n \\end{cases}
\n \\]
\n 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\n \\[\n \\begin{cases}\n \\gamma(0)=q\\\\\n \\dot\\gamma(0)=v.\n \\end{cases}\n \\]\nThis derive a map $\\exp_p$ form a neighborhood of $T_pM$ to $M$, and in fact it is a locally diffeomorphism, called \\emph{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 \\emph{geodesic polar coordinates}.\n \n 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\n \\begin{align}\n 0&=\\nabla_{\\dot\\gamma(t)}\\dot\\gamma(t)\n =\\nabla_{\\dot\\gamma(t)}\\left(\\dot r(t)\\pr\\right)+\n \\nabla_{\\dot\\gamma(t)}\\left(\\dot\\theta(t)\\pt\\right)\\notag\\\\\n &=\\dt\\dot r(t)\\cdot\\pr+\\dot r(t)\\cdot\\nabla_{\\dot\\gamma(t)}\\prr\n +\\dt\\dot\\theta(t)\\cdot\\pt+\\dot\\theta(t)\\cdot\\nabla_{\\dot\\gamma(t)}\\ptt\\notag\\\\\n &=\\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\\\\\n &\\qquad+\\dot\\theta(t)\\left(\\dot r(t)\\nabla_{\\pr}\\ptt+\\dot\\theta(t) \\nabla_{\\pt}\\ptt\\right)\\notag\\\\\n &=\\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}.\n \\end{align}\n The last equation obtained, since the $r$-curves are geodesics, thus $\\nabla_{\\prr}\\prr=0$ and note also that $[\\prr,\\ptt]=0$, hence $\\nabla_{\\ptt}\\prr=\\nabla_{\\prr}\\ptt$.\n \n the \\emph{Gauss lemms} assert that $\\langle\\prr,\\ptt\\rangle=0$, then the metric of $M$ can be written as\n \\begin{equation}\\label{eq:2}\n (\\mathrm ds)^2=(\\mathrm d r)^2+f^2(r,\\theta)(\\mathrm d\\theta)^2,\n \\end{equation}\n where $f(r,\\theta)$ is a positive function. Using the \\emph{compatibility} and \\emph{torsion free} again, we have \n \\begin{alignat*}{2}\n \\langle\\nabla_{\\ptt}\\prr,\\prr\\rangle&=0,\\quad&\\langle\\nabla_{\\ptt}\\prr,\\ptt\\rangle&=\\langle\\nabla_{\\prr}\\ptt,\\ptt\\rangle=ff_r,\\\\\n \\langle\\nabla_{\\ptt}\\ptt,\\prr\\rangle&=-ff_r,\\quad&\n \\langle\\nabla_{\\ptt}\\ptt,\\ptt\\rangle&=ff_\\theta.\n \\end{alignat*}\n Inserting these relations into \\eqref{eq:1},\n \\[\n 0=\\left\\{\\ddot r(t)-\\dot\\theta(t)\\dot\\theta(t)ff_r\\right\\}\\pr+\n \\left\\{\\ddot\\theta(t)+2\\dot r(t)\\dot\\theta(t)ff_r+\\dot\\theta(t)\\dot\\theta(t)ff_\\theta\\right\\}\\pt.\n \\]\n Thus, the geodesic equation is\n \\begin{equation}\\label{eq:3}\\begin{cases}\n 0&=\\ddot r(t)-\\dot\\theta(t)\\dot\\theta(t)ff_r\\\\\n 0&=\\ddot\\theta(t)+2\\dot r(t)\\dot\\theta(t)ff_r+\\dot\\theta(t)\\dot\\theta(t)ff_\\theta,\n \\end{cases}\\end{equation}\n where the $f,f_r$ should be evaulated at $\\gamma(t)=(r(t),\\theta(t))$.\n \n 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 \\emph{method moving frame}.\n \n Set $\\omega^1=\\rd r$, $\\omega^2=f\\rd\\theta$, then \\eqref{eq:2} can be re-written as\n \\[\n (\\rd s)^2=\\omega^1\\omega^1+\\omega^2\\omega^2.\n \\]\n A directly calculation will show that\n \\[\\begin{cases}\n \\rd\\omega^1&=\\rd(\\rd r)=0\\\\\n \\rd\\omega^2&=\\rd(f\\rd\\theta)=f_r\\rd r\\wedge\\rd\\theta,\n \\end{cases}\\]\n thus, if we set $\\omega_1^2=f_r\\rd\\theta=-\\omega_2^1$, then\n \\[\\begin{cases}\n \\rd\\omega^1&=\\omega^2\\wedge\\omega_2^1\\\\\n \\rd\\omega^2&=\\omega^1\\wedge\\omega_1^2\uff0c\n \\end{cases}\\]\n thus the \\emph{Cartan's Lemma} asserts that $\\omega^2_1$ is the connection 1-form and the \\emph{Gauss equation} says that\n \\[\n \\rd\\omega_1^2=-K\\omega^1\\wedge\\omega^2,\n \\]\n where $K$ is the Gauss curvature of $M$, which is a constant by assumption.\n \n 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\n \\begin{equation}\\label{eq:4}\n f_{rr}+K f=0.\n \\end{equation}\n Hence, since $t$ is arc-length parameter, the geodesic equations are\n \\begin{equation}\n \\begin{cases}\n 0&=\\ddot r(t)-\\dot\\theta(t)\\dot\\theta(t)ff_r\\\\\n 0&=\\ddot\\theta(t)+2\\dot r(t)\\dot\\theta(t)ff_r+\\dot\\theta(t)\\dot\\theta(t)ff_\\theta\\\\\n 1&=\\dot r(t)^2+f^2\\dot\\theta(t)^2\\\\\n 0&=f_{rr}+K f.\n \\end{cases}\n \\end{equation}\n<\/p>\n","protected":false},"excerpt":{"rendered":"
$\\newcommand{\\pr}{\\left.\\frac{\\partial}{ …<\/p>\n