---- > [!definition] Definition. ([[geodesic on a Riemannian manifold]]) > Let $M$ be a [[Riemannian manifold|Riemannian]] [[smooth manifold|manifold]]. > A [[parameterized curve|curve]] $\gamma(t):I \subset \mathbb{R} \to M$ is said to be a **geodesic in $M$** if its [[canonical lift to the tangent bundle|canonical]] [[lifting|lift]] $\gamma_{E}(t)$ to the [[tangent bundle]] $TM$ is a [[horizontal lift with respect to a connection on a vector bundle|horizontal lift]] [[connection on a manifold|with]] [[connection on a vector bundle|respect to]] the [[Levi-Civita connection]] on $M$. > > > ```tikz > \usepackage{tikz-cd} > \usepackage{amsmath} > \begin{document} > \begin{tikzcd} > & \text{Geodesic} & \\ > \begin{aligned} > &\text{Horiz. lift to } E \\ > &\text{w.r.t. conn. } A > \end{aligned} > \arrow[ru, "{E:= TM, A:=\text{L.C.}}" description] > & > & \begin{aligned} > &\text{Canon. lift} \\ > &\text{to } TM > \end{aligned} > \arrow[lu] > \end{tikzcd} > > \end{document} > ``` > > In [[coordinate chart|local coordinates]], we can state this condition explicitly by substituting $\dot{x}^{j}$ for $a^{j}$ in the [[system of ODEs|system]] of [[ODE|ODEs]] $\ref{{\dagger}}$ [[horizontal lift with respect to a connection on a vector bundle|defining]] a [[horizontal subspace|horizontal]] [[lifting|lift]]: locally, a geodesic is a solution to the second-order system of ODEs (the **geodesic equation**) $\begin{align} 0 &= \ddot{x}^{i} + \Gamma^{i}_{jk}(x) \dot{x}^{j} \dot{x}^{k} \\ i,j,k& = 1, \dots, \text{dim }M. \end{align}$ (This system is nonlinear.) Given any initial conditions (initial position and velocity) $\begin{cases}x(0) =p \in M \\\dot{x}(0) = a \in T_{p}M\end{cases}$ there is a unique solution for $|t|<\varepsilon$ for some $\varepsilon>0$ dependent on $p,a$. Denote this geodesic $\gamma_{p}(t, a)$. > Note that from ODE theory the result depends smoothly on the choice of initial conditions, that is, $\gamma$ is $C^{\infty}$ as a function of $(p,a) \in M \times TM$. ^definition > [!equivalence] > In coordinate-free language, the geodesic equation is $\nabla_{\dot{\gamma}}\dot{\gamma}=0.$ That is, the following are equivalent: > > 1. $\gamma$ is a geodesic > 2. $\gamma$ has zero acceleration > 3. $\gammas own velocity field $\dot{\gamma}$ is [[parallel section|parallel]] along $\gamma$ > > It is a bit like we are integrating the parallel transport equation twice. > Indeed, plugging $\dot{\gamma}$ for $X$ in the local [[parallel section|parallel transport]] equation, we get $\Gamma^{i}_{jk} \dot{\gamma}^{j}\dot{\gamma}^{k}+ \ddot{\gamma}^{i}(t)=0,$ which matches the local equation above. $\dot{\gamma}=\gamma_{*}\left( \frac{d}{dt} \right)$ $\frac{d}{dt} g(\dot{\gamma}, \dot{\gamma})=\dot{\gamma}g(\dot{\gamma}, \dot{\gamma})=2g(\nabla_{\dot{\gamma}} \dot{\gamma}, \dot{\gamma})=0$ the first equality is kind of chain rule, kind of just notation... it's all a bit messy > [!basicproperties] > - (Zero [[acceleration vector of a parameterized curve|acceleration]]) $\gamma$ is a geodesic if and only if the [[covariant derivative along a curve|covariant derivative along]] $\gamma$, $D_{\dot{\gamma}}\dot{\gamma}$, is zero for all $t$, just as a unit-parameterized [[line]] in $\mathbb{R}^{n}$ has zero acceleration. (This follows directly from the considerations in [[covariant derivative along a curve]].) >- (Constant [[velocity vector of a parameterized curve|speed]]) $|\dot{\gamma}(t)|_{g}\equiv \text{const.}$ for any geodesic $\gamma:I \to M$. So any solution of the geodesic equation is a curve parameterized with constant speed, and in particular will be a regular curve unless that speed is zero (in which case it is the trivial curve that stays still for all time). >- (Symmetric affine invariance) If $\gamma_{p}(t, a)$ solves the geodesic equation, then so do $\gamma_{p}(\lambda t, a)$ and $\gamma_{p}(t, \lambda a)$ for any $\lambda \in \mathbb{R}$. In fact, these are the same solution up to a reparameterization: $\gamma(\lambda t, a)=\gamma(t, \lambda a).$ > ^properties **Proof of constant speed.** This is an application of [[covariant derivative]] and [[Levi-Civita connection]] properties, [[covariant derivative along a curve|now that we know how to make sense of]] the notation $D_{\dot{\gamma}}\dot{\gamma}$ and know that the geodesic assumption means $D_{\dot{\gamma}}\dot{\gamma}=0$. In particular, the discussion in that note says we can wrap $\dot{\gamma}(t)$ in an ambient [[vector field]] $X$, and therefore make sense of things like '$\dot{\gamma}(t)$ acting as a [[derivation]]'. Have $\dot{\gamma} \cdot g(\dot{\gamma}, \dot{\gamma})=g(D_{\dot{\gamma}}\dot{\gamma}, \gamma)+ g(\dot{\gamma}, D_{\dot{\gamma}}\dot{\gamma})$ by definition (orthogonality) of [[Levi-Civita connection]]. The RHS vanishes because geodesics have zero acceleration. Can show that $\frac{d}{dt} |\dot{\gamma}(t)|_{g}^{2}=\dot{x}^{i} \frac{ \partial }{ \partial x^{i} } g(\dot{\gamma}, \dot{\gamma})$, meaning the [[derivative]] of the function $\|\dot{\gamma}(t)\|_{g}^{2}: \mathbb{R} \to \mathbb{R}$ is zero. So $|\dot{\gamma}(t)|$ is constant. We have $\begin{align} \frac{d}{dt} \|\dot{\gamma}(t)\|_{g}^{2} &= \frac{d}{dt} g({\dot\gamma, \dot{\gamma}}) \\ &= \dot{ \gamma} \cdot g(\dot{\gamma}, \dot{\gamma}) \\ &= g(\underbrace{ D_{\dot{\gamma}}\dot{\gamma} }_{ 0 }, \dot{\gamma})+ g(\dot{\gamma}, \underbrace{ D_{\dot{\gamma}}\dot{\gamma} }_{ 0 }) \\ &= 0. \end{align}$ ---- #### [[tangent space at a point of a smooth manifold|tangent space]] ---- #### References > [!backlink] > ```dataview > TABLE rows.file.link as "Further Reading" > FROM [[]] > FLATTEN file.tags as Tag > WHERE Tag = "#definition" OR Tag = "#theorem" OR Tag = "#MOC" OR Tag = "#proposition" OR Tag = "#axiom" > GROUP BY Tag > ``` > [!frontlink] > ```dataview > TABLE rows.file.link as "Further Reading" > FROM outgoing([[]]) > FLATTEN file.tags as Tag > WHERE Tag = "#definition" OR Tag = "#theorem" OR Tag = "#MOC" OR Tag = "#proposition" OR Tag = "#axiom" > GROUP BY Tag > ```