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Mathematics Qualifying Exam Question Bank

Department of Mathematics

Question

Math 6510 • Differentiable Manifolds

Fall 2002 • Question 10

Question

Math 6510 • Differentiable Manifolds Fall 2002 • Question 10

Define the index of a vector field on a smooth manifold. Construct an explicit vector field on the 2-sphere \(S^2\) and compute the Euler characteristic of \(S^2\) by computing the index of this vector field.

Raw LaTeX

Define the index of a vector field on a smooth manifold. Construct an explicit
vector field on the 2-sphere \(S^2\) and compute the Euler characteristic of \(S^2\) by
computing the index of this vector field.

Topic Tags

Euler Characteristic and Lefschetz Fixed Point Theorem Left-Invariant Vector Fields Smooth Manifolds and Embedded Submanifolds Vector Fields and Flows