4. Let M" and N be smooth manifolds, TM and TN be their tangent bundles....
50.1K
Verified Solution
Link Copied!
Question
Accounting
4. Let M" and N be smooth manifolds, TM and TN be their tangent bundles. (a) Define what is a smooth vector field on M. (b) Let $2n-1 denote the (2n 1)-dimensional unit sphere in R2n for n > 2. Show that its tangent bundle TS2n-1 admits at least one smooth nowhere vanishing vector field X. (You may wish to try n = 2 first.) (C) Suppose TM and TN are both trivial. Show that T(MXN) is trivial. (Recall an l-dimensional smooth manifold P' has trivial tangent bundle TP if P admits / linearly independent smooth vector fields X1, ... , X1. Note that an example is M = N = St so that M N = T2, the torus.) 4. Let M" and N be smooth manifolds, TM and TN be their tangent bundles. (a) Define what is a smooth vector field on M. (b) Let $2n-1 denote the (2n 1)-dimensional unit sphere in R2n for n > 2. Show that its tangent bundle TS2n-1 admits at least one smooth nowhere vanishing vector field X. (You may wish to try n = 2 first.) (C) Suppose TM and TN are both trivial. Show that T(MXN) is trivial. (Recall an l-dimensional smooth manifold P' has trivial tangent bundle TP if P admits / linearly independent smooth vector fields X1, ... , X1. Note that an example is M = N = St so that M N = T2, the torus.)
Answer & Explanation
Solved by verified expert
Get Answers to Unlimited Questions
Join us to gain access to millions of questions and expert answers. Enjoy exclusive benefits tailored just for you!
Membership Benefits:
Unlimited Question Access with detailed Answers
Zin AI - 3 Million Words
10 Dall-E 3 Images
20 Plot Generations
Conversation with Dialogue Memory
No Ads, Ever!
Access to Our Best AI Platform: Flex AI - Your personal assistant for all your inquiries!
