Topology on manifold and metric

[paweld]

Is there any relation between topology on manifold (which comes from
$$\mathbb{R}^n$$) and topology induced form metric in case
of Remanian manifold. What if we consider pseudoremaninan manifold.

 

[dodelson]

The Riemannian (or semi-Riemannian) metric doesn't directly induce a topology on the manifold itself, because it's an inner product on each of the tangent spaces, i.e. a norm TpM cross TpM -> R. Having said that, there is a natural topology inherited from the metric geodesics on a Riemannian manifold, where the distance between two points is the length of the shortest geodesic between them. This topology does turn out to be the same as the standard one.

-Matt

 

[JustinLevy]

Having said that, there is a natural topology inherited from the metric geodesics on a Riemannian manifold, where the distance between two points is the length of the shortest geodesic between them. This topology does turn out to be the same as the standard one.

When Penrose applied topology in GR to prove some singularity theorems, wasn't this referring to the topology of the metric on the manifold and not the topology of the manifold itself?

I may be greatly misusing words, so please even just restate definitions if I'm misunderstanding on a really basic level. But I basically considered Penrose and Hawking talking about the "causal topology" (I'm not sure what to call it), ie the connection of what events can in principle affect other events. This seems like something the manifold cannot have on its own. This is a topology "of the metric"?

At the very least, when people refer to topological arguments in GR, they usually aren't referring to just the topology of the manifold, correct?

 

[dodelson]

I don't think you're misusing words, it's just that the word topology is used for lots of different things. It seemed to me that the OP was asking about the point-set topology of the manifold. This means that we need to determine which subsets can be considered open. Since a manifold is locally Euclidean, the most natural thing to do is to just give it the open sets of Euclidean space. Penrose and Hawking do this as well.

The singularity theorems deal with a different kind of topology, the causal structure of the manifold. You're right; this can only be determined from the metric. For instance, we can define the causal past J-(S) of a set S as all points that can send out causal, or nowhere spacelike, curves to S. This depends on the metric, as being spacelike is a property of the norm of a tangent vector.

A third kind of topology that people consider is the global topology of a manifold, like how many holes it has or what its (co)homology and homotopy groups look like. This is still done using the standard open sets inherited from Rn.

-Matt

 

[paweld]

The singularity theorems deal with a different kind of topology, the causal structure of the manifold. You're right; this can only be determined from the metric.

Can we define topology induced from Lorentz metric? (metric needs to be
positive definite in order to induce a topology)