I Trying to construct a particular manifold locally using a metric

[dsaun777]

I am trying to construct a particular manifold locally using a metric, Can I simply take the inner product of my basis vectors to first achieve some metric.

 

[fresh_42]

What do you have and where do you want to go to? Where are your basis vectors from? What is the space?

Your question is far too vague to be answered.

 

[dsaun777]

What do you have and where do you want to go to? Where are your basis vectors from? What is the space?

Your question is far too vague to be answered.

Let me shorten it a little further, can I take the inner product of basis vectors and form a suitable metric?

 

[fresh_42]

Yes, if the inner product is one, i.e. positive definite, then it defines angles, lengths and distance. However, manifolds are usually not flat, which is why I wondered what your inner product is.

 

[dsaun777]

Yes, if the inner product is one, i.e. positive definite, then it defines angles, lengths and distance. However, manifolds are usually not flat, which is why I wondered what your inner product is.

Inner product is scalar

 

[Orodruin]

Let me shorten it a little further, can I take the inner product of basis vectors and form a suitable metric?

Sorry, but it is completely unclear what you mean by this. You have still not answered the question in #2: What do you have and where do you want to go?

 

[dsaun777]

Sorry, but it is completely unclear what you mean by this. You have still not answered the question in #2: What do you have and where do you want to go?

Assume I am in some surface of a differentiable manifold embedded in a 3 dimensional space. Given that I have some covariant basis vectors along the surface, is it as simple as taking the inner product of basis along the surface to construct a metric? How do you describe the surface in terms of the ambient 3d space using normal surface vectors?

 

[fresh_42]

You don't have covariant basis vectors of the surface, only of its tangent space which is a different one at each point. If you want to transport the metric onto the surface, you can either use the outer metric, since it is an embedded surface, or line integrals on the surface. See also connections. The outer metric is more or less worthless, as it does not measure an actual distance, which would be a geodesic.

 

[dsaun777]

You don't have covariant basis vectors of the surface, only of its tangent space which is a different one at each point. If you want to transport the metric onto the surface, you can either use the outer metric, since it is an embedded surface, or line integrals on the surface. See also connections. The outer metric is more or less worthless, as it does not measure an actual distance, which would be a geodesic.

Well forget about the ambient space. What would be the best way of finding a metric if you could only work with the basis vectors of the manifold?

 

[fresh_42]

This depends on what you want to do. Normally, it is
https://en.wikipedia.org/wiki/Riemannian_manifold#Riemannian_manifolds_as_metric_spaces