# Is the Levi-Civita connection unique for a given manifold?

#### meldraft

Hi all,

To my understanding, the Levi-Civita connection is the torsion-free connection on the tangent bundle preserving a given Riemannian metric.

Furthermore, given any affine connection ∇, there is a unique torsion-free connection ∇′ with the same family of affinely parametrized geodesics. (from Wikipedia)

Does this imply that for a given Riemannian manifold there is a Levi-Civita connection that is unique for that manifold?

I am still learning this stuff so I am often having trouble seeing whether something is true or not :)

Edit: To elaborate, from the fundamental theorem of riemannian geometry, there is a unique torsion-free metric connection, called the Levi-Civita connection of the given metric.

Does this make this connection unique for that manifold? i.e. if I have a lot of manifolds can I identify them using the Levi-Civita connection?

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#### quasar987

Homework Helper
Gold Member
There is one Levi-Civita connection for each metric. But even different metrics can have the same Levi-Civita connection: for any positive constant c, the two metrics g and cg have the same Levi-Civita connection because it is easy to see that the Levi-Civita connection of g also is compatible with cg.

#### meldraft

Of course, this makes sense, thank you!

On a side (or maybe main)-note, do you know if there is some other quantity that uniquely defines the manifold?

In my mind, if you start deforming a sphere, you can end up with infinite unique shapes, but the deviation from the original curvature can be calculated by the curvature tensor field.

Is there some way to define the manifold without having to calculate the curvature tensor all over the manifold? (I want to do it numerically)

I am sorry if I am asking something that makes no sense, but as I said, this is pretty new to me :)

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#### quasar987

Homework Helper
Gold Member
I don't theink that there is such a quantity, but there are interesting and deep classification theorems. See for instance the Uniformization theorem on wiki. According to the book by John Lee, much current research in riemannian geometry is devoted to "extending" the uniformization theorem to higher dimensions.

#### meldraft

This is very helpful, thank you!

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