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Construct a smooth vector field on the sphere S2 that vanishes only at one point

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HOMEWORK 3 MATH-GA 2350.001 DIFFERENTIAL GEOMETRY I (due by October, 17, 2016)

1. Consider the usual charts on RP2:

[x, y, z]7→(u1, u2) = (x/z, y/z)on Uz ={z6= 0};

[x, y, z]7→(v1, v2) = (x/y, z/y) onUy ={y6= 0};

[x, y, z]7→(w1, w2) = (y/x, z/x) onUx ={x6= 0}.

Show that there is a vector field on RP2 which in the last coordinate chart has the following expression:

w1 ∂

∂w1

−w2 ∂

∂w2

.

What are the expressions of this vector field in the other two charts?

2. Construct a smooth vector field on the sphere S2 that vanishes only at one point.

3. Let M, N be smooth manifolds and let f :M →N be a smooth submersion.

Show that for any smooth vector field Y on N there exists a smooth vector field X onM such that

∀x∈M Txf(X(x)) =Y(f(x)).

1

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