What does it mean by a Riemannian metric on a vector bundle?

In summary: On each fiber ,it is just a metric on the fiber viewed as a vector space.Different fibers are different vector spaces and generally have different metrics - which means there is no natural way to compare these vector spaces or their metrics.Thanks!
  • #1
petergreat
267
4
It's really a question about convention. Does such a metric have to be linear on each fiber?
 
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  • #2
petergreat said:
It's really a question about convention. Does such a metric have to be linear on each fiber?

symmetric bilinear form on each fiber
 
  • #3
lavinia said:
symmetric bilinear form on each fiber

Does it have to preserve the natural Euclidean metric up to a constant factor in each fiber (which is a vector space)?
 
  • #4
In other words, are we allowed to "curve" the base space only or the entire space?
 
  • #5
petergreat said:
Does it have to preserve the natural Euclidean metric up to a constant factor in each fiber (which is a vector space)?

not sure what you mean but each fiber is a vector space with a metric defined on it. Different fibers have there own separate metric and there is generally no way to compare them among different fibers.

There is generally no natural Euclidean metric on a fiber.

If you have a submanifold of another manifold then its tangent and normal bundles inherit a metric from the metric on the tangent space of the ambient manifold.
 
  • #6
lavinia said:
not sure what you mean but each fiber is a vector space with a metric defined on it. Different fibers have there own separate metric and there is generally no way to compare them among different fibers.

There is generally no natural Euclidean metric on a fiber.

If you have a submanifold of another manifold then its tangent and normal bundles inherit a metric from the ambient manifold.

I'm talking about a vector bundle, so each fiber has a natural metric up to constant factor.
 
  • #7
petergreat said:
I'm talking about a vector bundle, so each fiber has a natural metric up to constant factor.

no. There is no natural metric. Why do you think that? Can you give me a proof?
 
  • #8
lavinia said:
no. There is no natural metric. Why do you think that? Can you give me a proof?

Oops... You're right. But still, does it have to be a constant 2-tensor on each fiber?
 
  • #9
petergreat said:
Oops... You're right. But still, does it have to be a constant 2-tensor on each fiber?

On each fiber ,it is just a metric on the fiber viewed as a vector space.

Different fibers are different vector spaces and generally have different metrics - which means there is no natural way to compare these vector spaces or their metrics.
 
  • #10
Thanks! That's clear now.
 

1. What is a Riemannian metric?

A Riemannian metric is a mathematical concept that assigns a notion of length and angle to a given space. It is a way to measure distances and angles in a space, similar to how a ruler or protractor is used in geometry.

2. What is a vector bundle?

A vector bundle is a mathematical structure that consists of a space (called the base space) and a collection of vector spaces (called the fibers) attached to each point in the base space. It is a generalization of the concept of a vector field, where the vector spaces can vary smoothly over the base space.

3. How is a Riemannian metric defined on a vector bundle?

A Riemannian metric on a vector bundle is defined as a smooth assignment of a Riemannian metric to each fiber of the bundle. This means that at each point in the base space, there is a Riemannian metric defined on the corresponding fiber. This allows for a consistent way to measure distances and angles in the vector bundle.

4. What is the significance of a Riemannian metric on a vector bundle?

A Riemannian metric on a vector bundle is important because it enables us to define geometric notions such as length, angle, curvature, and volume in a vector bundle. This is useful in various areas of mathematics and physics, such as differential geometry, general relativity, and gauge theory.

5. How does a Riemannian metric on a vector bundle relate to the underlying base space?

A Riemannian metric on a vector bundle is closely related to the Riemannian metric on the base space. In fact, the Riemannian metric on the base space can be viewed as a special case of a Riemannian metric on a vector bundle where the fibers are all one-dimensional (i.e. just a single vector space). This relationship allows for the study of geometric properties of the base space using tools and concepts from vector bundle theory.

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