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

  • Context: Graduate 
  • Thread starter Thread starter petergreat
  • Start date Start date
  • Tags Tags
    Mean Metric Vector
Click For Summary
SUMMARY

A Riemannian metric on a vector bundle does not require linearity on each fiber, as each fiber is treated as a separate vector space with its own distinct metric. There is no natural Euclidean metric on a fiber, and different fibers cannot be directly compared due to their unique metrics. The discussion emphasizes that while a fiber may have a natural metric up to a constant factor, this does not imply a uniform metric across fibers. Additionally, metrics can be inherited from the ambient manifold when dealing with submanifolds.

PREREQUISITES
  • Understanding of Riemannian geometry
  • Familiarity with vector bundles
  • Knowledge of fiber spaces and their properties
  • Concept of metrics in differential geometry
NEXT STEPS
  • Study the properties of Riemannian metrics in vector bundles
  • Explore the concept of fiber metrics and their implications
  • Learn about the inheritance of metrics from ambient manifolds
  • Investigate the differences between various types of vector spaces in differential geometry
USEFUL FOR

Mathematicians, differential geometers, and students of geometry interested in the properties of vector bundles and Riemannian metrics.

petergreat
Messages
266
Reaction score
4
It's really a question about convention. Does such a metric have to be linear on each fiber?
 
Physics news on Phys.org
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.
 

Similar threads

Undergrad Pseudo-Riemaniann isometries
  • · Replies 4 ·
Replies
4
Views
2K
Undergrad Identification tangent bundle over affine space with product bundle
  • · Replies 13 ·
Replies
13
Views
3K
Graduate On the dependence of the curvature tensor on the metric
  • · Replies 6 ·
Replies
6
Views
3K
Undergrad The second fundamental form and derived metrics
  • · Replies 13 ·
Replies
13
Views
3K
Undergrad About the maximal extension of local charts on a manifold
  • · Replies 8 ·
Replies
8
Views
1K
Undergrad Lie derivative of a metric determinant
  • · Replies 20 ·
Replies
20
Views
7K
Graduate Riemannian Metric Tensor & Christoffel Symbols: Learn on R2
  • · Replies 7 ·
Replies
7
Views
2K
Undergrad Galilean spacetime as a fiber bundle
  • · Replies 16 ·
Replies
16
Views
5K
Undergrad Confused about the inner product
  • · Replies 58 ·
2
Replies
58
Views
5K
Undergrad Is There a Distinction Between Riemann Metrics in Physics?
  • · Replies 16 ·
Replies
16
Views
3K