Geodesics: From Affine Connection to Length Minimisation

On a manifold $$M$$ with affine connection $$\nabla$$, I can define a geodesic as a special integral curve. Namely a geodesic is an integral curve $$\gamma$$, generated by a vector field $$X$$, that also parallel transports this tangent vector i.e. it satisfies

$$\tag{\star}\nabla_v v =0$$

Note that we are able to define the geodesic without any mention of a metric at all

Q1: does this mean geodesics exist on non-Riemannian manifolds?

Q2: how can I start with the definition $$(\star)$$, introduce a metric tensor $$g$$ and show that the geodesic will be the curve that minimises arclength. Everything I read assumes that the connection is the affine connection. I am aware that the metric picks out the affine connection as the only metric compatible connection (satisfying $$\nabla g=0$$) but that doesn't mean other connections don't exist. Supposing $$(\star)$$ was defined with respect to a different connection, what happens?

• The definition that I heard of was that $\gamma$ is a geodesic on a smooth manifold $M$ with connection $\nabla$ iff the covariant derivative of $\gamma'$ is $0$: $$D_t \gamma'\equiv0.$$ No metric is needed here 🙂. But it seems clear that if you choose different metrics $g$, then $\gamma$ can't be minimizing for all of them. Jun 21, 2020 at 21:23
• For Q2: Not every affine connection preserves a Riemannian metric, so you cannot go from geodesics defined by (*) to Riemannian geodesics. Jun 21, 2020 at 23:21
• For any metric $g$, there are many connections $\nabla$ satisfying $\nabla g = 0$. If $\nabla$ is any such connection (e.g., the Levi-Civita connection) then $\nabla'_X Y := \nabla_X Y + D(X, Y)$ for any tensorial map $D : TM \times TM \to TM$ satisfying $D(Y, X) = -D(X, Y)$. But among these only the Levi-Civita is torsion-free, so the relevant uniqueness statement is that for any metric $g$ there is a unique torsion-free connection compatible with $g$, and by definition this is the Levi-Civita connection. Jun 23, 2020 at 16:51

Q1. Like you say, any connection $$\nabla$$ on a smooth manifold $$M$$ determines a set of geodesics without any need for a metric. We call the structure $$(M, \nabla)$$ an affine manifold.

Q2. For most connections $$\nabla$$ there is no metric $$g$$ whose geodesics coincide with those $$\nabla$$.

On the other hand, for any connection $$\nabla$$ there is a unique torsion-free connection $$\nabla'$$ with the same geodesics, so we may as well restrict our attention to torsion-free connections. (In terms of the Christoffel symbols, the new connection is given by $$(\Gamma')_{ab}^c = \frac{1}{2}(\Gamma_{ab}^c + \Gamma_{ba}^c$$).)

Now, any connection $$\nabla$$ is specified locally by its Christoffel symbols, and for a torsion-free connection $$\nabla$$, we have $$\Gamma_{ba}^c = \Gamma_{ab}^c$$, so a connection is given in local coordinates by $$\frac{1}{2} n^2 (n + 1)$$ functions, where $$n := \dim M$$. But a metric is specified in local coordinates by $$\frac{1}{2} n (n + 1)$$ functions, so, informally, for $$n > 1$$ there are many more connections than metrics.

Put another way, the map $$\mathcal C : \{\textrm{metrics on M}\} \to \{\textrm{torsion-free affine connections on M}\}$$ that assigns to a metric $$g$$ on $$M$$ its Levi-Civita connection $$\nabla^g$$ is not surjective. In fact, it is not injective either; for a typical Levi-Civita connection $$\nabla^g$$ the only metrics whose geodesics are those of $$\nabla^g$$ are those homothetic to $$g$$, that is, the metrics $$\lambda g$$, $$\lambda > 0$$, but for some metrics there are others (e.g., all of the metrics $$g_{ij} \, dx^i \,dx^j$$ on $$\Bbb R^n$$ with $$g_{ij}$$ constant have the same geodesics as the standard Euclidean metric, $$g_{ij} = \delta_{ij}$$).

Remark One can ask how to determine for a given torsion-free connection $$\nabla$$ whether it is the Levi-Civita connection of some metric. A partial answer is provided by various tensorial obstructions to metrizability, that is, tensors defined invariantly in terms of $$\nabla$$ that vanish if $$\nabla$$ is a Levi-Civita connection. The simplest of these is the trace $$Q_{ab} := R_{ab}{}^c{}_c \in \Gamma(\bigwedge^2 T^* M)$$ of the curvature over the last two indices, that is, the section $$Q(X, Y) = \operatorname{tr}(Z \mapsto R(X, Y) Z) = \sum_{i=1}^n e^i (R(X, Y) E_i),$$ where $$(E_i)$$ is some local frame and $$(e^i)$$ is its dual coframe. This quantity vanishes iff $$\nabla$$ (locally) preserves some volume form---and any Levi-Civita connection $$\nabla^g$$ preserves any local volume form for $$g$$---but a generic connection has $$Q \neq 0$$ and so preserves no volume form locally. This obstruction is not sharp, that is, there are connections for which $$Q = 0$$ but which are not Levi-Civita connections. One can construct other, more sophisticated (and sensitive) obstructions.

• Right. One can add that in the simply-connected case, the Ambrose-Singer theorem computes the Lie algebra of the holonomy group (in terms of the curvature of the connection). A connection admits a compatible Riemannian metric iff the holonomy is compact, and one can detect compactness of a Lie algebra using the Killing form. Jun 23, 2020 at 22:14

You can use such definition without using a metric, for example see the notion of affine manifold, like the quotient of $$\mathbb{R}^n-\{0\}$$ by the homothetic map $$h(x)=2x$$, it is endowed with a connection inherited from the classical flat connection of $$\mathbb{R}^n-\{0\}$$ since that connection is preserved by $$h(x)=2x$$.

Geodesic can be defined in Riemannian geometry with the distance. A Riemannian metric on $$M$$ induces a distance and if $$M$$ is complete, a geodesic between $$x,y$$ with will be the path between $$x$$ and $$y$$ which is the critical point of function. See the answer here.

Shortest path to a geodesic

Given a metric $$g$$, the Levi-Civita connexion is the unique connexion which satisfy two further conditions : first, $$\nabla g =0$$, then its torsion is $$0$$.

These two condition are needed to prove that geodesics "minimize" distance in they satisfy Euler Lagrange equation for the Lagrangian $$\int g(\gamma '(t), \gamma '(t)) dt$$.