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

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Discussion Overview

The discussion revolves around the nature of a Riemannian metric on a vector bundle, focusing on whether such a metric must be linear on each fiber, how it relates to the Euclidean metric, and the implications of curving the base space versus the entire space.

Discussion Character

  • Debate/contested
  • Conceptual clarification

Main Points Raised

  • Some participants question whether a Riemannian metric must be linear on each fiber of the vector bundle.
  • There is a discussion about whether the metric has to preserve the natural Euclidean metric up to a constant factor in each fiber, with some arguing that there is generally no natural Euclidean metric on a fiber.
  • Participants explore the idea of curving the base space versus the entire space, raising questions about the implications for the metric.
  • It is noted that different fibers have their own separate metrics, and there is generally no way to compare these metrics among different fibers.
  • One participant suggests that each fiber has a natural metric up to a constant factor, while another challenges this assertion, asking for proof.
  • There is a clarification that on each fiber, the metric is simply a metric on the fiber viewed as a vector space, and different fibers are different vector spaces with generally different metrics.

Areas of Agreement / Disagreement

Participants do not reach consensus on whether a Riemannian metric must be linear on each fiber or whether there is a natural metric on the fibers. Multiple competing views remain regarding the nature of metrics on vector bundles.

Contextual Notes

Participants express uncertainty about the definitions and assumptions regarding metrics on fibers and the comparison of different fibers' metrics.

petergreat
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It's really a question about convention. Does such a metric have to be linear on each fiber?
 
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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
 
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)?
 
In other words, are we allowed to "curve" the base space only or the entire space?
 
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.
 
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.
 
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?
 
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?
 
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.
 

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