Connections in non-Riemannian geometry

Question

In case of Riemannian geometry the connection $\Gamma^i_{jk}$ as is derived from the derivatives of the metric tensor $g_{ij}$ is ought to be symmetric wrt to its lower two indices. But in the case of Non-Riemannian Geometry that need not be the case, so the question is how do you actually construct such connections? Do you again use the metric tensor?

Answer 1

Here's one way to construct a connection on the tangent bundle (a similar construction works on more general vector bundles). Let $\{\rho_\alpha \}$ be a partition of unity subordinate to a locally finite coordinate cover $\{U_\alpha \}$. On each $U_\alpha$, choose coordinates $x_1^\alpha, \dots, x_n^\alpha$, giving a frame $X_1^\alpha = \frac{\partial}{\partial x_1^\alpha}, \dots , X_n^\alpha = \frac{\partial}{\partial x_n^\alpha}$ for the tangent bundle. Then choose a connection $\nabla^\alpha$ on each $U_\alpha$; any valid connection will do! To specify $\nabla^\alpha$, it suffices to specify the Christoffel symbols. One easy choice would be to make all the Christoffel symbols zero (meaning $\nabla^\alpha_{X_i^\alpha} X_j^\alpha = 0$ for all $i$, $j$).

We now have a connection on each $U_\alpha$, but we don't have a well-defined connection on the whole manifold because in general, $\nabla^\alpha$ and $\nabla^\beta$ will not agree on $U_\alpha \cap U_\beta$. But one way to construct a global connection is to use the partition of unity: define $\nabla$ by $$ \nabla := \sum_\alpha \rho_\alpha \nabla^\alpha.$$ Then $\nabla$ is a well-defined global connection. This shows that connections exist!

Of course, depending on the context, this construction may not be too useful since we chose the $\nabla^\alpha$'s arbitrarily. For example, on a Riemannian manifold, one usually wants to work with the unique Levi-Civita connection. On more general vector bundles, there is not a canonical connection analogous to the Levi-Civita connection, but one often wants to work with metric-compatible connections (the torsion-free condition does not make sense in general).

Answer 2

A connection is an abstract non-unique $\mathbb{R}$-linear map on a smooth manifold $M$ with tangent bundle $TM$ and smooth vector fields $\Gamma(TM)$ given by $$\nabla:\begin{cases}\Gamma(TM)\times \Gamma(TM)\to \Gamma(TM)\\ (X,Y)\mapsto \nabla_XY\end{cases}$$ satisfying the following properties

1. $\nabla_{fX}Y=f\nabla_XY$ for all $f\in C^\infty(M)$
2. $\nabla_X(fY)=X(f)Y+f\nabla_XY$
3. $\nabla_XY-\nabla_YX=[X,Y]$

It is a fact that the zero map doesn't satify above properties, but that if one has two connections $\nabla^1,\nabla^2$ one can create another by $g\nabla^1+(1-g)\nabla^2$. When looking at Riemannian manifolds $(M,g)$ we get a unique connection called Levi-Civita Connection which in addition satisfies what one calles metric compatibility $$Xg(Y,Z)=g(\nabla_XY,Z)+g(Y,\nabla_XZ)$$ The proof mainly consists in looking at all cyclic permutations of above equations and than using a smart way in combining them to get an explicit formula which shows existance and uniqueness and satisfies the above properties. This formula is $$2g(\nabla_XY,Z)=Xg(Y,Z)+Yg(Z,X)-Zg(X,Y)+g([X,Y],Z)-g([X,Z],Y)-g([Y,Z],X)$$

What you denoted as connection is in fact not a connection but the so called Christoffel Symbols obtained from the Levi-Civita connection via $$\nabla_{\partial_j}\partial_k=\sum\limits_{j,k}\Gamma^i_{jk}\partial_i$$ where $\partial_i,\partial_j,\partial_k$ are defined by taking a chart $(U,,\varphi)$ at a point $p\in M$ and setting $$\partial_i(p)(f)=\frac{\partial (f\circ\varphi^{-1})}{\partial x_i}(\varphi(p))$$ This is used to express a vector field in local coordinates $X(p)=\sum\limits_iX_i(p)\partial_i(p)$ and gives an expression of the Levi-Civita connection in local coordinates $$\nabla_XY=\sum\limits_{i=1}^m\left(\sum_jX_j\partial_jY_i+\sum\limits_{j,k}\Gamma_{jk}^iX_jY_k\right)\partial_i$$ Like this we realize, that the Christoffel Symbol can be interpret as a measure of how strong the derivative of a vector field along another vector field on a Riemannian manifold deviates from the standart partial derivative on $\mathbb{R}^n$.