Could anyone please help me on the following problem. Please write down your thoughts. Thanks alot. amd939 said:
A parallelizable manifold is a type of smooth manifold that has a smooth vector field defined on it, which means that at every point on the manifold, there is a tangent vector that is linearly independent from the other tangent vectors at that point. This allows for parallel transport of vectors along any path on the manifold.
Examples of parallelizable manifolds include Euclidean space, spheres, tori, and projective spaces. In fact, any Lie group is also a parallelizable manifold.
Parallelizable manifolds are important in differential geometry and topology because they allow for the construction of non-trivial vector bundles. They also play a key role in the study of characteristic classes and other topological invariants of manifolds.
In physics, parallelizable manifolds are used to model physical systems that possess symmetries. For example, the spacetime in Einstein's theory of general relativity is a parallelizable manifold, allowing for the concept of parallel transport of vectors along geodesics.
No, not every smooth manifold is a parallelizable manifold. In fact, it has been proven that there are only a finite number of parallelizable manifolds in each dimension. For example, there are only two parallelizable manifolds in dimension 2, the 2-sphere and the real projective plane.