Geodesics and Completeness

Geodesics

Let $(M,g)$ be a smooth Riemannian manifold. Let $\gamma :[0,1]\to M$ be a smooth curve connecting two fixed points $p,q\in M$ such that: $\gamma (0)=p,\gamma (1)=q$ The Length Functional $L(\gamma )$ of the curve is defined by integrating its speed with respect to the Riemannian metric $g$: $L(\gamma ):={\int }_0^1\Vert \dot{\gamma }(t){\Vert }_{g}dt={\int }_0^1(g_{\gamma (t)}(\dot{\gamma }(t),\dot{\gamma }(t)){)}^{1/2}dt$ where $\dot{\gamma }(t)\in T_{\gamma (t)}M$ is the velocity vector field along the curve. A curve $\gamma :[a,b]\to M$ in a Riemannian manifold $(M,g)$ is called a geodesic if it is a critical point of the length functional.

In the calculus of variations, the critical points of the length functional $L(\gamma )$ coincide with the critical points of the Energy Functional $E(\gamma )={\int }_0^1\Vert \dot{\gamma }(t){\Vert }_g^{2}dt$, provided the curve is parametrized with constant speed. Working with $L(\gamma )$ directly introduces a scaling factor of $\frac{1}{\Vert \dot{\gamma }(t)\Vert }$ in the derivation.

Proposition

Let $(M,g)$ be a Riemannian manifold, and let $\mathcal{P}(p,q)$ be the space of smooth curves $\gamma :[a,b]\to M$ with fixed endpoints $\gamma (a)=p$ and $\gamma (b)=q$. For any parameterized curve $\gamma \in \mathcal{P}(p,q)$ with $\dot{\gamma }(t)\ne 0$, the following statements are equivalent:

1. $\gamma$ satisfies the differential equation of a geodesic: ${\nabla }_{\dot{\gamma }(t)}\dot{\gamma }(t)=0\forall t\in [a,b]$
2. $\gamma$ is a critical point of the energy functional: $E(\gamma )={\int }_a^b\Vert \dot{\gamma }(t){\Vert }_g^{2}dt$
3. $\gamma$ is a critical point of the length functional: $L(\gamma )={\int }_a^b\Vert \dot{\gamma }(t){\Vert }_{g}dt$ and $\gamma$ is parameterized with constant speed (i.e., $\Vert \dot{\gamma }(t){\Vert }_g=c$ for some constant $c\ge 0$).

To find the paths that minimize (or extremize) $L(\gamma )$, we introduce a smooth proper variation. This is a parameterized family of curves $\gamma (s,t):(-ϵ,ϵ)\times [0,1]\to M$ satisfying:

1. $\gamma (0,t)=\gamma (t)$ (the original curve).
2. $\gamma (s,0)=p$ and $\gamma (s,1)=q$ for all $s$ (fixed endpoints). Infinitesimally, this variation defines a variation vector field $V(t)$ along $\gamma (t)$, which belongs to the space of sections $\Gamma ([0,1],{\gamma }^{\ast }TM)$: $V(t):=\frac{\partial \gamma }{\partial s}(0,t)\in T_{\gamma (t)}M$ Because the endpoints are fixed for all $s$, the variation vector field vanishes at the boundaries: $V(0)=0,V(1)=0$ We now compute the first variation of the length functional with respect to the parameter $s$ at $s=0$: $\frac{d}{ds}{|}_{s=0}L(\gamma (s,t))={\int }_0^1\frac{d}{ds}{|}_{s=0}⟨\frac{\partial \gamma }{\partial t},\frac{\partial \gamma }{\partial t}{⟩}^{1/2}dt$ Using the chain rule and the metric compatibility of the Levi-Civita connection $\nabla$, we can differentiate under the inner product: $={\int }_0^1\frac{1}{2\Vert \dot{\gamma }(t)\Vert }\cdot 2⟨\frac{D}{ds}{|}_{s=0}\frac{\partial \gamma }{\partial t},\frac{\partial \gamma }{\partial t}⟩dt$ $={\int }_0^1\frac{1}{\Vert \dot{\gamma }(t)\Vert }⟨\frac{D}{ds}{|}_{s=0}\frac{\partial \gamma }{\partial t},\dot{\gamma }(t)⟩dt$ Because the Levi-Civita connection is torsion-free, the coordinate derivatives commute when acting on a parameterized surface $\gamma (s,t)$. This gives the crucial identity: $\frac{D}{ds}\frac{\partial \gamma }{\partial t}=\frac{D}{dt}\frac{\partial \gamma }{\partial s}$ Evaluating this at $s=0$ switches the derivatives, allowing us to express the integrand in terms of our variation vector field $V(t)$: $\frac{D}{ds}{|}_{s=0}\frac{\partial \gamma }{\partial t}=\frac{D}{dt}V(t)={\nabla }_{\dot{\gamma }}V(t)$ Substituting this back into our integral yields: $\frac{d}{ds}{|}_{s=0}L(\gamma (s,t))={\int }_0^1\frac{1}{\Vert \dot{\gamma }(t)\Vert }⟨{\nabla }_{\dot{\gamma }}V(t),\dot{\gamma }(t)⟩dt$ To isolate $V(t)$, we apply the product rule (metric compatibility of $\nabla$): $\frac{d}{dt}⟨V(t),\dot{\gamma }(t)⟩=⟨{\nabla }_{\dot{\gamma }}V(t),\dot{\gamma }(t)⟩+⟨V(t),{\nabla }_{\dot{\gamma }}\dot{\gamma }(t)⟩$ Rearranging for the term in our integrand: $⟨{\nabla }_{\dot{\gamma }}V(t),\dot{\gamma }(t)⟩=\frac{d}{dt}⟨V(t),\dot{\gamma }(t)⟩-⟨{\nabla }_{\dot{\gamma }}\dot{\gamma }(t),V(t)⟩$ Plugging this back into the first variation formula: $\frac{d}{ds}{|}_{s=0}L(\gamma (s,t))={\int }_0^1\frac{1}{\Vert \dot{\gamma }(t)\Vert }\left(\frac{d}{dt}⟨V(t),\dot{\gamma }(t)⟩-⟨{\nabla }_{\dot{\gamma }}\dot{\gamma }(t),V(t)⟩\right)dt$ The first term is a total derivative. Since $V(0)=V(1)=0$, integrating it yields zero via the Fundamental Theorem of Calculus. This leaves: $\frac{d}{ds}{|}_{s=0}L(\gamma (s,t))=-{\int }_0^1\frac{1}{\Vert \dot{\gamma }(t)\Vert }⟨{\nabla }_{\dot{\gamma }(t)}\dot{\gamma }(t),V(t)⟩dt$

By the Fundamental Lemma of the Calculus of Variations, a curve $\gamma (t)$ is a critical point of the length functional if and only if the first variation vanishes for all valid choice of variation fields $V(t)$. $\frac{d}{ds}{|}_{s=0}L(\gamma (s,t))=0\forall V(t)⟺{\nabla }_{\dot{\gamma }(t)}\dot{\gamma }(t)=0$ This is the Geodesic Equation written in coordinate-free notation. Remark (Constant Speed): Any curve satisfying the geodesic equation automatically has a constant speed. We can check this by differentiating its squared norm: $\frac{d}{dt}\Vert \dot{\gamma }(t){\Vert }^2=2⟨{\nabla }_{\dot{\gamma }(t)}\dot{\gamma }(t),\dot{\gamma }(t)⟩=2⟨0,\dot{\gamma }(t)⟩=0$ To express this system as ordinary differential equations (ODEs), choose a local coordinate chart $(U,\varphi )$ such that the curve is represented by component functions $\gamma (t)=(x^1(t),x^2(t),\dots ,x^m(t))$. The velocity vector field expands in terms of the coordinate frame ${\partial }_i=\frac{\partial }{\partial x^i}$ as: $\dot{\gamma }(t)={\sum }_{i=1}^m{\dot{x}}^i(t){\partial }_i.$Substituting this into the geodesic equation ${\nabla }_{\dot{\gamma }}\dot{\gamma }=0$ and applying the Leibniz rule yields: ${\nabla }_{\dot{\gamma }}\left({\sum }_{i=1}^m{\dot{x}}^i{\partial }_i\right)={\sum }_{i=1}^m{\ddot{x}}^i{\partial }_i+{\sum }_{i=1}^m{\dot{x}}^i{\nabla }_{\dot{\gamma }}{\partial }_i=0$ Expanding the directional covariant derivative ${\nabla }_{\dot{\gamma }}{\partial }_i$ using the chain rule along the path: ${\nabla }_{\dot{\gamma }}{\partial }_i={\nabla }_{\left(\sum {\dot{x}}^j{\partial }_j\right)}{\partial }_i={\sum }_{j=1}^m{\dot{x}}^j{\nabla }_{{\partial }_j}{\partial }_i$ Recall that the Christoffel symbols ${\Gamma }_{ji}^k$ characterize the connection on coordinate fields: ${\nabla }_{{\partial }_j}{\partial }_i={\sum }_{k=1}^m{\Gamma }_{ji}^k{\partial }_k$. Substituting this back gives: ${\sum }_{k=1}^m{\ddot{x}}^k{\partial }_k+{\sum }_{i=1}^m{\sum }_{j=1}^m{\sum }_{k=1}^m{\dot{x}}^i{\dot{x}}^j{\Gamma }_{ji}^k{\partial }_k=0$ Factoring out the basis vectors ${\partial }_k$ and noting their linear independence, each component must vanish independently. This delivers the classical Geodesic ODE System: ${\ddot{x}}^k(t)+{\sum }_{i,j=1}^m{\Gamma }_{ij}^k(\gamma (t)){\dot{x}}^i(t){\dot{x}}^j(t)=0,\forall k=1,\dots ,m$

Riemannian Exponential Map

Let $(M,g)$ be a semi-Riemannian manifold, and fix a point $p\in M$. The Riemannian exponential map ${\exp }_p:T_{p}M\to M$ sends a tangent vector $v$ to the point reached by following the geodesic $\gamma$ starting at $p$ with initial velocity $v$, for a unit time. That is ${\exp }_p(v):={\gamma }_v(1).$

Remark: The name here exponential map corresponds to the exponential map in Lie group and Lie Algebra.

Geodesic completeness

We call a manifold geodesic completeness if $\forall p\in M$,

1. ${\exp }_p:T_{p}M⟶M$ is well-defined for all vectors in $T_{p}M$ $v⟼{\exp }_p(v).$
2. The geodesic $\gamma (t)$ on $M$ is well-defined for all $t\in \mathbb{R}$. $|\dot{\gamma }(t)|\text{ is constant in }t.$ By the uniqueness of the ODE on $TM$: ${\exp }_p(t\cdot v)={\exp }_p\left(c\cdot t\cdot \frac{v}{c}\right)$

The Hopf–Rinow Theorem

Let $(M,g)$ be a connected Riemannian manifold, then the following are equivalent:

1. $(M,g)$ is geodesically complete.
2. $(M,d_g)$ is a complete metric space.
3. For some $p\in M$, the exponential map ${\exp }_p:T_{p}M⟶M$ is well-defined for all $v\in T_{p}M$.
4. Any two points $p$ & $q$ of $M$ can be joined by a length-minimizing geodesic.
5. A closed & bounded subset of $M$ is compact.

Lemma

Let $\gamma :[0,a)⟶M$ be a geodesic. If ${\lim }_{t\to a}\gamma (t)\text{ exists in }M,$ then the geodesic $\gamma (t)$ can be extended beyond $t=a$.

Lemma

If $(M,g)$ is geodesically complete, then any 2 points $p$ & $q$ can be joined by a length-minimizing geodesic.