Why are there infinitely many connections on a Riemannian manifold? I've just started learning some Riemannian manifold stuff, and I'm getting confused about the concept of connection. A connection $\nabla: \Gamma(T\mathcal{M})\times \Gamma(T\mathcal{M}) \rightarrow \Gamma(T\mathcal{M})$ basically defines rules of differentiation $\nabla_XY$ on the tensor field of a manifold, but how can there be infinitely many connections? Does it mean that we may define arbitrary differentiation rules (as long as they satisfy the linear and product axioms) on a tensor field? Of course, one special connection is the Levi-Civita connection, but I don't see how we may arbitrarily define $\Gamma_{i,j}^k$ to generate different connections.
Also, isn't the covariant derivative defined by projecting the usual directional derivative onto the same tangent space? If so it seems the rule of differentiation on a vector field can already be determined, so why there exist other forms of connections?
Thanks!
 A: I'm turning my comment into an answer because it got too long.
You may want to look at this question and answer, in which I give a way of constructing a connection. The many choices involved in the construction should make it clear that many connections exist.
When you say "projecting the usual directional derivative onto the tangent space", I assume you are defining a connection on a submanifold $M$ of $\mathbb{R}^n$, in which case there is a natural Riemannian metric $g$ on $M$ (induced from the ambient metric on $\mathbb{R}^n$). One can show the connection you describe is the Levi-Civita connection associated to $g$, and so it's the natural connection in that context. But there exist other connections, and of course if we put a different metric on $M$ (e.g., associated to a different embedding), we would get a different Levi-Civita connection.
When I first learned about connections, like you, I wondered what the point of considering connections other than the Levi-Civita connection was. I now realize that on a Riemannian manifold, people almost always seem to use the Levi-Civita connection on the tangent bundle. But on vector bundles other than the tangent bundle, there does not necessarily exist a canonical choice of connection analogous to the Levi-Civita connection, and there are often many connections that one could choose to get the job done (whatever that job may be).
