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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!

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  • $\begingroup$ This answer might be helpful to gain more intuition. $\endgroup$ – guy-in-seoul Sep 10 '14 at 0:53
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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).

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  • $\begingroup$ Thanks for your answer! But why can we specify $\Gamma_{i,j}^k$ freely? Since $\Gamma_{i,j}^k$ determines $\nabla_{E_i}E_j$, does it mean that we may define the covariant derivatives of frames arbitrarily? $\endgroup$ – thinkbear Sep 10 '14 at 15:36
  • $\begingroup$ I think after all we are free to determine the differentiation rules as long as the axioms are satisfied, and this allows defining arbitrary linear maps between two adjacent tangent spaces, as well as their corresponding $\Gamma_{i,j}^k$. Levi-Civita connection corresponds to the map that preserves inner products, whose $\Gamma_{i,j}^k$ turns out to be a function of $g_{i,j}$. Hopefully this understanding is correct. $\endgroup$ – thinkbear Sep 10 '14 at 22:52
  • $\begingroup$ You can almost specify the $\Gamma$'s freely, except that you have to make sure that the connection you construct is well-defined (i.e., independent of the choice of coordinates on overlaps of coordinate charts). This is why I used a partition of unity in the answer I linked to...But locally at least, yes, you can basically specify the $\Gamma$'s arbitrarily. $\endgroup$ – Phillip Andreae Sep 12 '14 at 3:34

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