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HOMEWORK 4 MATH-GA 2350.001 DIFFERENTIAL GEOMETRY I (due by October, 24, 2016) 1. Show that on R

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HOMEWORK 4 MATH-GA 2350.001 DIFFERENTIAL GEOMETRY I (due by October, 24, 2016)

1. Show that onR² the vector fields y²_∂x^∂ andx²_∂y^∂ are complete, but the vector field y²_∂x^∂ +x²_∂y^∂ is not complete.

2. Let a∈Rⁿ. Let X_a be the vector field defined byX_a(y) =a−y.

(a) Compute the flow of X_a;

(b) If b∈X_b, compute [X_a, X_b]using the flows of X_a and X_b.

3. Find global noncommuting flows on S².

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