## Question

1) Show that s1 + s2 is a smooth section of E.

2) Let Γ(E) = C∞(E) be the set of all smooth sections of E. Define the multiplication of sections from Γ(E) by real numbers in a natural way. Is Γ(E) an R-linear space?

Problem 2. How to define a structure of C∞(X)-module on Γ(E)?

Problem 3. Let π1 : E1 → X and π2 : E2 → X be two smooth vector bundles over X. How to define their direct sum and their tensor product as smooth vector bundles.

Problem 4. Suppose there are given transition functions {φij} and {ψij} for the bundles π1 : E1 → X and π2 : E2 → X. What are the transition functions for the vector bundles E1 ⊕ E2, E1 ⊗ E2, E ͮ1.

## Solution Preview

These solutions may offer step-by-step problem-solving explanations or good writing examples that include modern styles of formatting and construction of bibliographies out of text citations and references. Students may use these solutions for personal skill-building and practice. Unethical use is strictly forbidden.

By purchasing this solution you'll be able to access the following files:

Solution.pdf.