[Tu Differential Geometry] 14. Geodesics

 This article is one of Manifold, Differential Geometry, Fibre Bundle.

In this section we prove the existence and uniqueness of a geodesic with a specified initial point and a specified initial velocity. These geodesics played a pivotal role in the histoy of non-Euclidean geometry, for they proved for the first time that the parallet postulate is independent of Euclid's first four postulates.

14.1 The Definition of a Geodesic

A staright line in $\mathbb{R}^n$ with parametrization $$c(t)=p+tv,\quad p,v\in \mathbb{R}^n$$ is characterized by the property that its acceleration $c''(t)$ is identically zero. If $T(t)=c'(t)$ is the tangent vector of the curve at $c(t)$, then $c''(t)$ is also covariant derivative $DT/dt$ associated to the Euclidean connection $D$ on $\mathbb{R}^n$. This example suggests a way to generalize the notion of "straightness" to an arbitrary manifold with a connection.

Definition 14.1. Let $M$ be a manifold with an affine connection $\nabla$ and $I$ an open, closed, or half-open interval in $\mathbb{R}$. A parametrized curve $c:I\rightarrow M$ is geodesic if the covariant derivative $DT/dt$ of its velocity vector filed $T(t)=c'(t)$ is zero. The geodesic is said to be maximal if its domain $I$ cannot be extended to a larger interval.

Remark 14.2. The notion of a geodesic depends only on a connection and does not require a metric on the manifold $M$. However, if $M$ is a Riemannian manifold, then we will always take the connection to be the unique Riemannian connection. On a Riemannian manifold, the speed of a curve $c(t)$ is defined to be the magnigude of its velocity vector: $$\lVert c'(t)\rVert = \sqrt{\left\langle c'(t),c'(t)\right\rangle}.$$

Proposition 14.3. The speed of a geodesic on a Riemannian manifold is constant.

Proof. Let $T=c'(t)$ be the velocity of the geodesic. The speed is constant if and only if its square $f(t)=\langle T,T\rangle$ is constant. But $$f'(t)=\frac{d}{dt}\langle T,T\rangle = 2\left\langle \frac{DT}{dt},T\right\rangle=0.$$ So $f(t)$ is constant. 

Let $M$ be a smooth surface in $\mathbb{R}^3$. Recall that its Riemannian connection is given by $$\nabla_XY=(D_XY)_\mbox{tan},$$ the tangential component of the directional derivative $D_XY$. If $N$ is a unit normal vector field on an open subset $U$ of $M$, then the tangential component of a vector field $Z$ along $M$ is $$Z_\mbox{tan}=Z-\langle Z,N\rangle N.$$ For a vector field $V$ along a curve $c(t)$ in $M$, the covariant derivative associated to the Riemannian connection $\nabla$ on $M$ is $$\frac{DV}{dt}=\left( \frac{dV}{dt}\right)_\mbox{tan}.$$

14.2 Reparametrization of a Geodesic

Proposition 14.5 (Linear Function Reparametrization). Suppose $\gamma(u)$ is a nonconstant geodesic on a manifold with a connection and $\bar{\gamma}(t):=\gamma(u(t))$ is a reparametrization of $\gamma(u)$. Then $\bar{\gamma}(t)$ is a geodesic if and only if $u=\alpha t+\beta$ for some real constants $\alpha$ and $\beta$.

Proof. Let $T(u)=\gamma'(u)$ and $\bar{T}(t)=\bar{\gamma}'(t)$ be the tangent vector fields of the two curves $\gamma(u)$ and $\bar{\gamma}(t)$. By the chain rule, $$\bar{T}(t)=\frac{d}{dt}\gamma(u(t))=\gamma'(u(t))u'(t)=u'(t)T(u(t)).$$ By property (ii) of a covariant derivative, $$\frac{D\bar{T}}{dt}=u''(t)T(u(t))+u'(t)\frac{DT(u(t))}{dt}\\ =u''(t)T(u(t))+u'(t)\frac{DT}{du}\frac{du}{dt}=u''(t)T(u(t)),$$ where $DT/du=0$ because $\gamma(u)$ is a geodesic. Since $T(u)$ has constant length, it is never zero. Therefore, $$\frac{D\bar{T}}{dt}=0\iff u''=0 \iff u=\alpha t+\beta$$ for some $\alpha,\beta\in \mathbb{R}$.

Corollary 14.6. Let $(a,b)$ be an interval containing $0$. For any positive constant $k\in\mathbb{R}^+$, the curve $\gamma(u)$ is a geodesic on $(a,b)$ with initial point $q$ and initial vector $v$ if and only if $\bar{\gamma}(t):=\gamma(kt)$ is a geodesic on $(a/k.b/k)$ with initial point $q$ and initial vector $kv$.

Proof. Since $\frac{a}{k}<t<\frac{b}{k}\iff a<kt<b,$ the curve $\bar{\gamma}(t):=\gamma(kt)$ is defined on $(a/k,b/k)$ if and only if $\gamma(t)$ is defined on $(a,b)$. By the chain rule, $$\begin{align} \bar{\gamma}(t)=k\gamma'(kt).\end{align}$$ Equivalently, $$\bar{T}(t)=kT(kt),$$ so that $$\frac{D\bar{T}}{dt}(t)=k^2\frac{DT}{dt}(kt).$$ Thus, $\bar{\gamma}$ is a geodesic if and only if $\gamma$ is a geodesic. They have  the same initial point $\bar{\gamma}(0)=\gamma(0)$. By (1), their initial vectors are related by $$\bar{\gamma}'(0)=k\gamma'(0).$$

14.3 Existence of Geodesics

Suppose $M$ is a manifold with a connection $\nabla$, and $(U,\phi)=(U,x^1,\cdots,x^n)$ is a chart relative to which the Christoffel symbols are $\Gamma^k_{ij}$. Let $c:[a,b]\rightarrow M$ be a smooth curve. In the chart $(U,\phi)$ the curve $c$ has coordinates $$(\phi\circ c)(t)=(x^1(c(t)),\cdots,x^n(c(t))).$$ We will write $y(t)=(\phi\circ c)(t)$ and $y^i(t)=(x^i\circ c)(t)$, and say that $c$ is given in $U$ by $$y(t)=(y^1(t),\cdots,y^n(t)).$$ In this section we will determine a set of differential equations on $y^i(t)$ for the curve to be a geodesic.

Denote by $\dot{y}^i$ the first derivative $dy^i/dt$ and by $\ddot{y}^i$ the second derivative $d^2y^i/dt^2$. Let $\partial_i=\partial/\partial x^i$. Then $$T(t)=c'(t)=\sum \dot{y}^i \partial_{j,c(t)}$$ and $$\begin{align} \frac{DT}{dt}(t)=\sum_j \ddot{y}^j\partial_{j,c(t)}+\sum_j\dot{y}^j\nabla_{c'(t)}\partial_j=\sum_j \ddot{y}^j\partial_{j,c(t)}+\sum_{i,j}\dot{y}^j\nabla_{\dot{y}^i\partial_{i,c(t)}}\partial_j.\end{align}$$ To simplify the notation, we sometimes write $\partial_j$ to mean $\partial_{j,c(t)}$ and $\Gamma^k_{ij}$ to mean $\Gamma^k_{ij}(c(t))$. Then (2) becomes  $$\frac{DT}{dt}=\sum_j\ddot{y}^j\partial_j+\sum_{i,j}\dot{y}^j\nabla_{\dot{y}^i\partial_i}\partial_j=\sum_k \ddot{y}^k\partial_k+\sum_{i,j,k}\dot{y}^i\dot{y}^j\Gamma^k_{ij}\partial_k=\sum_k \left( \ddot{y}^k + \sum_{i,j} \dot{y}^i\dot{y}^j\Gamma^k_{ij}\right) \partial_k.$$ So $c(t)$ is a geodesic if and only if $$\ddot{y}^k+\sum_{i,j}\Gamma^k_{ij}\dot{y}^i\dot{y}^j=0,\quad k=1,\cdots,n.$$ We summaroze tjos doscissopm om tje following theorem.

Theorem 14.7 (Coordinate form of Geodesic Equation). On a manifold with a connection, a parametrized curve $c(t)$ is a geodesic if and only if relative to any chart $(U,\phi)=(U,x^1,\cdots, x^n)$, the components of $(\phi\circ c)(t)=(y^1(t),\cdots,y^n(t))$ satisfy the system of differential equations $$\begin{align} \ddot{y}^k+\sum_{i,j}\Gamma^k_{ij}\dot{y}^i\dot{y}^j=0,\quad k=1,\cdots,n,\end{align}$$ where the $\Gamma^k_{ij}$'s are evaluated on $c(t)$.

This is a system of second-order ordinary differential equations on the real line, called the geodesic equations. By an existence and uniqueness theorem of ordinary differential equations, we have the following theorem.

Theorem 14.8 (Existence and Uniqueness of Geodesic). Let $M$ be a manifold with a connection $\nabla$. Given any point $p\in M$ and tangent vector $v\in T_pM$, there is a geodesic $c(t)$ in $M$ with initial point $c(0)=p$ and initial velocity $c'(0)=v$. Moreover, this geodesic is unique in the sense that any other geodesic satisfying the same initial conditions must agree with $c(t)$ on the intersection of their domains.

Let $\gamma_v(t,q)$ be the unique maximal geodesic with initial point $q$ and initial vector $v\in T_qM$. We also write $\gamma_v(t)$. By Corollary 14.6, $$\gamma_v(kt)=\gamma_{kv}(t)$$ for any positive real number $k$.

On a Riemannian manifold we always use the unique Riemannian connection to define geodesics. In this case, tangent vectors have lengths and the theory of ordinary differential equations gives the following theorem.

Theorem 14.10 (Existence and Uniqueness of Geodesic in Riemannian Manifold). For any point $p$ of a Riemannian manfiold $M$, there are a neighborhood $U$ of $p$ and numbers $\delta,a>0$ so that for any $q\in U$ and $v\in T_qM$ with $\lVert v\rVert <\delta$, there is a unique geodesic $\gamma:(-a,a)\rightarrow M$ with $\gamma(0)=q$ and $\gamma'(0)=v$.

We can rescale interval.

Thoerem 14.11 (Shorter domain). For any point $p$ of a Riemannian manifold $M$, there are a neighborhood $U$ of $p$ and a real number $\epsilon>0$ so that for any $q\in U$ and $\bar{v}\in T_qM$ with $\lVert \bar{v}\rVert <\epsilon$, there is a unique geodesic $\bar{\gamma}:(-2,2)\rightarrow M$ with $\bar{\gamma}(0)=q$ and $\bar{\gamma}'(0)=\bar{v}$.

Proof. Fix $p\in M$ and find a neighborhood $U$ of $p$ and positive numbers $\delta$ and $a$ as in Theorem 14.10. Set $k=a/2$. By Corollary 14.6, $\gamma(t)$ is a geodesic on $(-a,a)$ with initial point $q$ and initial vector $v$ if and only if $\bar{\gamma}(t):=\gamma(kt)$ is a geodesic on $(-a/k,a/k)=(-2,2)$ with initial point $q$ and initial vector $\bar{v}:=kv$. Moreover, $$\lVert v\rVert =\lVert \gamma'(0)\rVert <\delta \iff \lVert \bar{v} \rVert =\lVert \bar{\gamma}'(0)\rVert = k\lVert v\rVert <k\delta=\frac{a\delta}{2}.$$ Choose $\epsilon=a\delta/2$. Then Theorem 14.10 translates into this one.

Proposition 14.12. A connection-preserving differomorphism $f:(M,\nabla)\rightarrow (\tilde{M},\tilde{\nabla})$ takes geodesics to geodesics.

Proof. Suppose $c(t)$ is a geodesic in $M$, with tangent vector field $T(t)=c'(t)$. If $\tilde{T}(t)=(f\circ c)'(t)$ is the tangent vector field of $f\circ c$, then $$\tilde{T}(t)=f_*\left( c_*\frac{d}{dt}\right) =f_* (c'(t))=f_*T.$$ Let $D/dt$ and $tilde{D}/dt$ be the covariant derivatives along $c(t)$ and $(f\circ c)(t)$, respectively. Then $DT/dt\equiv 0$, because $c(t)$ is a geodesic. By Proposition 13.4, $$\frac{\tilde{D}\tilde{T}}{dt}=f_*\left( \frac{DT}{dt}\right) =f_*0=0.$$ Hence, $f\circ c$ is geodesic in $\tilde{M}$.

Because an isometry of Riemannian manifold preserves the Riemannian connection, by Proposition 14.12 it carries geodesics to geodesics.

14.4 Geodesics in the Poincare Half-Plane

circle

14.5 Parallel Translation

Closely related to geodesics is the notion of parallel translation along a curve. A parallel vector field along a curve is an analogue of a constant vector field in $\mathbb{R}^n$. Throughout the rest of this chapter we assume that $M$ is manifold with a connection $\nabla$ and $c:I\rightarrow M$ is a smooth curve in $M$ defined on some interval $I$.

Definition 14.13. A smooth vector field $V(t)$ along $c$ is parallel if $DV/dt\equiv 0$ on $I$.

In this terminology a geodesic is simply a curve $c$ whose tangent vector field $T(t)=c'(t)$ is parallel along $c$.

Fix a point $p=c(t_0)$ and a frame $e_1,\cdots,e_n$ in a neighborhood $U$ of $p$ in $M$. Let $[\omega^i_j]$ be the matrix of connection forms of $\nabla$ relative to this frame. If $V(t)=\sum v^i(t)e_{i,c(t)}$ for $c(t)\in U$, then $$\frac{DV}{dt}=\sum_i \dot{v}^ie_{i,c(t)}+\sum_i v^i\frac{De_{i,c(t)}}{dt}=\sum_i \dot{v}^ie_{i,c(t)}+\sum_jv^j\nabla_{c'(t)}e_j\\ =\sum_i \dot{v}^ie_{i,c(t)}+\sum_{i,j}v^j\omega^i_j(c'(t))e_{i,c(t)}=\sum_i \left( \dot{v^i}+\sum_j \omega^i_j(c'(t))v^j\right) e_{i,c(t)}.$$ Thus $DV/dt=0$ if and only if $$\begin{align}\dot{v^i}+\sum_j \omega^i_j(c'(t))v^j=0,\quad i=1,\cdots,n.\end{align}$$ This is a system of linear first-order ordinary differential equations. By the existence and uniqueness theorem of ordinary differential equations, there is always a unique solution $V(t)$ on a small interval about $t_0$ with a given $V(t_0)$. In the next subsection we show that in fact the solution exists not only over a small interval, but also over the entire curve $c$.

If $V(t)$ is parallel along a curve $c:[a,b]\rightarrow M$, then we say that $V(b)$ is obtained from $V(a)$ by parallel translation along $c$ or that $V(b)$ is the parallel translate or parallet transport of $V(a)$ along $c$. By the uniqueness theorem of ordinary differential equations, $V(t)$ is uniquely determined by the initial condition $V(a)$, so if parallel translation exists along $c$, then it is well defined.

14.6 Existence of Parallel Translation Along a Curve

While a geodesic is guaranteed to exist only locally, parallel translation is possible along the entire length of a curve.

Theorem 14.14 (Existence of Global Parallel Translation). Let $M$ be a manifold with a connection $\nabla$ and let $c:[a,b]\rightarrow M$ be a smooth curve in $M$. Parallel translation is possible from $c(a)$ to $c(b)$ along $c$, i.e., given a vector $v_0\in T_{c(a)}$, there exists a parallel vector field $V(t)$ along $c:[a,b]\rightarrow M$ such that $V(a)=v_0$. Parallel translation along $c$ induces a linear isomorphism $\varphi_{a,b}:T_{c(a)}(M)\rightarrow T_{c(b)}(M).$$

Proof. Because the parallel transport equation $DV/dt=0$ is $\mathbb{R}$-linear in $V$, a linear combination with constant coefficients of parallel vector fields along $c$ is again parallel along $c$.

Let $w_1,\cdots,w_n$ be a basis for $T_{c(t_0)}M$. For each $i=1,\cdots,n$, there is an interval inside $[a,b]$ of length $\epsilon_i$ about $t_0$ such that a parallel vector field $W_i(t)$ exists along $c:(t_0-\epsilon_i,t_0+\epsilon_i)\rightarrow M$ whose value at $t_0$ is $w_i$. For $\epsilon$ equal to the minimum of $\epsilon_1,\cdots,\epsilon_n$, the $n$ basis vectors $w_1,\cdots,w_n$ for $T_{c(t_0)}M$ can be parallel translated along $c$ over the interval $(t_0-\epsilon,t_0+\epsilon)$. By taking linear combinations as in the remark above, we can parallel translate every tangent vector in $T_{c(t_0)}M$ along $c$ over the interval $(t_0-\epsilon,t_0+\epsilon)$. 

For $t_1\in (t_0-\epsilon,t_0+\epsilon)$ parallel translation along $c$ produces a linear map $\varphi_{t_0,t_i}:T_{c(t_0)}M\rightarrow T_{c(t_1)}M$. Note that a vector field $V(t)$ is parallel along a curve $c(t)$ if and only if $V(-t)$ is parallel along $c(-t)$, the curve $c$ reparametrized with the opposite orientation. This shows that the linear map $\varphi_{t_0,t_1}$ has an inverse $\varphi_{t_1,t_0}$ and is therefore an isomorphism.

Thus for each $t\in [a,b]$ there is an open interval about $t$ over which parallel translation along $c$ is possible. Since $[a,b]$ is compact, it is covered by finitely many such open intervals. Hence, it is possible to parallel translate along $c$ from $c(a)$ to $c(b)$.

While a geodesic with a given initial point and initial velocity exists only locally, parallel translation is always possible along the entire length of a smooth curve. In fact, the curve need not even be smooth.

Definition 14.15. A curve $c:[a,b]\rightarrow M$ is piecewise smooth if there exist numbers $$a=t_0<t_1<\cdots,t_r=b$$ such that $c$ is smooth on $[t_i,t_{i+1}]$ for $i=0,\cdots,r-1$.

By parallel translating over each smooth segment in succession, one can parallel translate over any piecewise smooth curve.

14.7 Parallel Translation on a Riemannian Manifold

Parallel translation is defined on any manifold with a connection; it is not necessary to have a Riemannian metric. On a Riemannian manifold we will always assume that parallel translation is with respect to the Riemannian connection.

Proposition 14.16 (Parallel Transltation preserve Length and Inner Product). On a Riemannian manifold $M$ parallel translation preserves length and inner product: if $V(t)$ and $W(t)$ are parallel vector fields along a smooth curve $c:[a,b]\rightarrow M$, then the length $\lVert V(t)\rVert$ and the inner product $\langle V(t),W(t)\rangle$ are constant for all $t\in [a,b]$.

Proof. Since $\lVert V(t)\rVert =\sqrt{\langle V(t),V(t)\rangle}$, it suffices to prove that $\langle V(t),W(t)\rangle$ is constant. By the product rule for the covariant derivative of a connection compatible with the metric (Theorem 13.2), $$\frac{d}{dt}\langle V,W\rangle =\left\langle \frac{DV}{dt},W\right\rangle + \left\langle V,\frac{DW}{dt}\right\rangle =0,$$ since $DV/dt=0$ and $DW/dt=0$. Thus $\langle V(t),W(t)\rangle$ is constant as a function of $t$.