Is There a Limit to the Possible Metrics on a Riemannian Manifold?

[paweld]

I wonder how much information about metic tensor of Riemmanian manifold can be extracted if only the Levi-Civita connection is given.

Conversly, if the metric on manifold is given there is formula for Christoffel symbols which define connection so there exists only one symetrical metric connection on Riemmanian manifold. I presume that connection should strongly limited the class of possible metrics but I don't know how.

For example if Christoffel symbols in specific coordinate system vanish everywhere, the metric tensor in this base may be identity matrix times constant (are there any other possibilities??).

 

[Reb]

I wonder how much information about metic tensor of Riemmanian manifold can be extracted if only the Levi-Civita connection is given.

That's a bit funny to ask, since a Riemannian manifold must come along with a metric.

Conversly, if the metric on manifold is given there is formula for Christoffel symbols which define connection

Yes there is. That's what enables one to define the Levi-Civita connection in the first place. You can look it up in every good book about Differentiable Manifolds (like Do Carmo's book).

For example if Christoffel symbols in specific coordinate system vanish everywhere, the metric tensor in this base may be identity matrix times constant

I think everywhere vanishing Christoffel symbols means zero metric.

 

[wofsy]

That is a good question. I guess that multiplying the metric by a positive constant is the only possibility.

Compatibility says z.<x,y> = <DzY,Y> + <x,DzY>

If you multiply the metric by a non-constant function the equation can't work.

You get z.f<x,y> = f<DzY,Y> + f<x,DzY> + (x.f)<x,y>

I found it instructive to look at flat metrics that are then multiplied by a non-constant function. One gets geometries that are far from flat.
try taking a flat torus embedded in the 3 sphere (easy to write down) and projecting it into three space using 4 dimensional stereographic projection. This will give you a conformally flat torus. Its curvature is highly non-trivial. It looks like a slinky.

Another famous and important example is isothermal coordinates on surfaces. On every Riemannian surface there exists local coordinates which are conformal to the flat Euclidean metric i.e. the metric is a positive function multiplied by the standard Euclidean metric. So this shows that any geometry on a surface can be locally represented as a function times the standard flat metric.