Math 109b - Homework 2 Due: 27 January 2017 1. Show that if φ1 : U1 → S and φ2 : U2 → S are diffeomorphisms onto a domain U ⊂ S, then the area of φ1 and the area of φ2 are equal. 2. Let φ : S2 → R3 be the map given by φ : (x, y, z) 7→ (3x, 2y, z). Compute the first fundamental form and the area of φ. 3. Let γ : (−3π, π3 ) → S2 be the map given by  γ : t 7→

t t cos t, sin t, 3π 3π

r

 t2 1− 2 . 9π

Compute the length of γ. 4. Show that every smooth surface has a non-zero vector field. 5. Let S be a smooth surface and φ : S → R3 be a smooth immersion. Show that S is orientable if and only if there is a smooth vector field X along φ(S) so that • X(p) 6= 0 for all p ∈ S. • hX(p), V iφ(p) = 0 for all V ∈ φ∗,p (Tp S). Here, h·, ·iφ(p) is the standard inner product on Tφ(p) R3 . Hint: Use partitions of unity. 6. Use Problem 5to show that S2 is orientable, and that the M¨obius strip (0, 1) × [0, 1]) (y, 0) ∼ (1 − y, 1) is not orientable. 7. Show that there are only two (equivalence classes of) orientations on any connected orientable smooth surface. 8. Let φ : S1 → S2 be a diffeomorphism. (a) Prove that given an orientation on S1 , φ induces an orientation on S2 . (b) Supppose S1 = S2 = S. Is the orientation induced by φ and the orientation we started of with necessarily equivalent? If so, prove it, and if not, give an example.

Math 109b - Homework 2 Due: 27 January 2017

Jan 27, 2017 - Let φ : S2 → R3 be the map given by φ : (x, y, z) ↦→ (3x,2y, z). ... orientable if and only if there is a smooth vector field X along φ(S) so that.

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