1. Homework 11
(1) Let U = {(x, y) ∈ R2: y > 0}. Then U is an open subset of R2. Let p = (x0, y0) ∈ U be any point. On Tp(R2), we define
(1.1) h(a, b)p, (c, d)piTp(R2) = ac + bd y20 for any (a, b)p and (c, d)p in Tp(R2).
(a) Prove that h·, ·iTp(R2) defines an inner product on Tp(R2) for each p ∈ U.
(b) Let e1(p) and e2(p) be the vectors in Tp(R2) defined by v1(p) = (y0, 0)p, v2(p) = (0, y0)p.
Show that {v1(p), v2(p)} forms an orthonormal basis for Tp(R2) with respect to the inner product defined by (1.1).
(c) Let {θ1(p), θ2(p)} be the dual basis to the basis {v1(p), v2(p)} defined in Problem 1b. Express θ1(p) and θ2(p) in terms of {dxp, dyp}.
(d) We obtain two one forms {θ1, θ2} on U by considering p 7→ θi(p) for i = 1, 2.
Compute dθ1 and dθ2.
(e) Prove that there is a unique one form ω on U so that dθ1 = ω ∧ θ2, dθ2 = −ω ∧ θ1. Compute ω and dω.
(f) Since {θ1(p), θ2(p)} is a basis for Tp∗(R2) which is the dual basis to {e1(p), e2(p)}, in class, we learned that {θ1(p) ∧ θ2(p)} forms a basis for Λ2Tp∗(R2). Since ω is an one form, dω is a two form. Find the unique number K(p) so that
(dω)(p) = −K(p)(θ1(p) ∧ θ2(p)).
(2) Let (y1, y2, y3) be the coordinate functions on R3 and (x1, x2) be the coordinate functions on R2. Let f : R2→ R3 be a smooth function, where
f (x1, x2) = (f1(x1, x2), f2(x1, x2), f3(x1, x2)), (x1, x2) ∈ R3. (a) Express f∗dy1 and f∗dy2 and f∗dy3 in terms of dx1 and dx2. (b) Compute f∗(dyi∧ dyj) for 1 ≤ i < j ≤ 3.
(3) Let f : Rn→ Rm be a smooth function. Assume that η a smooth r-form and ω be a smooth s-form on Rm.
(a) Prove that ds(f∗ω) = f∗(dsω). Here dsω is the derivative of ω.
(b) Prove that f∗(ω ∧ η) = (f∗ω) ∧ (f∗η).
(4) Prove that dr(dr−1ω) = 0 for any smooth r − 1 form ω on Rn. (5) Let η a smooth r-form and ω be a smooth s-form on Rn. Prove that
dr+s(η ∧ ω) = (drη) ∧ ω + (−1)rη ∧ (dsω).
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