- #1
math6
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hello friends
my question is: if we have M a compact manifold, do we have there necessarily TM compact ?
thnx .
my question is: if we have M a compact manifold, do we have there necessarily TM compact ?
thnx .
mathwonk said:one way to make it compact is to replace TX by PTX, the bundle whose fibers are the projective spaces associated to the tangent vector spaces.
A compact manifold is a type of mathematical space that is both smooth and finite in size. It is a generalization of a surface in 3-dimensional space, and can have any number of dimensions. In simpler terms, it is a curved space that is finite and smooth.
Question 2:A tangent bundle is a mathematical construction that describes the set of all possible tangent vectors at every point on a manifold. It is essentially a collection of all the possible directions in which a manifold can be locally deformed.
Question 3:No, a compact manifold does not always imply a compact tangent bundle. In fact, there are examples of non-compact manifolds that have compact tangent bundles. This is because the compactness of a manifold only depends on its global structure, while the compactness of a tangent bundle depends on its local structure.
Question 4:A compact tangent bundle has important implications in the study of differential geometry and topology. It allows for the application of powerful mathematical techniques and theorems, and can provide insight into the global properties of a manifold. In essence, a compact tangent bundle simplifies the study of a manifold and allows for a deeper understanding of its structure.
Question 5:To prove that a compact manifold implies a compact tangent bundle, we can use the fact that a manifold is a locally compact space. This means that every point on a manifold has a compact neighborhood. By constructing a suitable partition of unity, we can show that the tangent bundle is also locally compact, and hence, compact. This proof technique is known as the 'Whitney embedding theorem'.