SO(n) actions on vector bundles

In summary, there is no natural action of SO(n) on the tangent bundle of an oriented Riemannian n-manifold or any oriented vector bundle with Riemannian metric over a smooth manifold. However, it does act on the frame bundle in a special case (2-dimensional oriented case). In general, the action depends on the choice of basis and the tangent vector being acted upon.
  • #1
wofsy
726
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An oriented surface with a Riemannian metric has a natural action of the unit circle on its tangent bundle. Rotate the tangent vector through the angle theta in the positively direction.

Is there a natural action of SO(n) on the tangent bundle of an oriented Riemannian n-manifold?

Same question for any oriented vector bundle with Riemannian metric over a smooth manifold.
 
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  • #2
In general, the answer is no. I don't see how one could define it independent of basis.

It does act, however, on the frame bundle (kind of like a lift of the tangent bundle to a principal bundle). The 2-dimensional case is quite special, since the frame bundle of an oriented 2-manifold is a circle (i.e. fix an angle - then orientation gives you an orthonormal frame).
 
  • #3
zhentil said:
In general, the answer is no. I don't see how one could define it independent of basis.

It does act, however, on the frame bundle (kind of like a lift of the tangent bundle to a principal bundle). The 2-dimensional case is quite special, since the frame bundle of an oriented 2-manifold is a circle (i.e. fix an angle - then orientation gives you an orthonormal frame).

Right - so what you are saying implies the following. take a unit vector and extend it to an oriented n-frame arbitrarily, keeping the vector as the first element of the frame. Let SO(n) act on this frame. Then the image of v under this action depends on the choice of the orthogonal complement to v.
 
  • #4
Yes, I believe that's precisely it. Unless the action fixes v (or v itself determines a basis, as in the 2-dimensional oriented case), you have to specify which tangent vector to send it to.
 

1. What is a SO(n) action on a vector bundle?

SO(n) refers to the special orthogonal group of n dimensions, which is a group of linear transformations that preserve the length of vectors. An SO(n) action on a vector bundle is a specific type of transformation that preserves the structure of the vector bundle, such as its fibers and base space, while also preserving the length of vectors.

2. How do SO(n) actions on vector bundles relate to differential geometry?

SO(n) actions on vector bundles are closely related to differential geometry because they allow for the study of geometric structures that are preserved under these transformations. This can be useful in understanding the curvature and other properties of the vector bundle.

3. What are some examples of SO(n) actions on vector bundles?

One example of an SO(n) action on a vector bundle is the rotation of a vector bundle on a sphere. Another example is the change of basis of a vector bundle, where the new basis is related to the original basis by an SO(n) matrix.

4. How are SO(n) actions on vector bundles used in physics?

SO(n) actions on vector bundles are used in physics to describe and study physical systems that exhibit symmetry under these transformations. This can include systems with rotational symmetry, such as celestial bodies, or systems with other types of symmetries that can be described by SO(n) actions.

5. What are some applications of understanding SO(n) actions on vector bundles?

Understanding SO(n) actions on vector bundles has numerous applications in mathematics and physics. It can be used to study symmetries in physical systems, to classify and analyze geometric structures, and to develop new mathematical tools and techniques for solving problems in these fields.

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