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!
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).