Math 225 B : Differential Geometry , Homework 4

Math 225 B : Differential Geometry , Homework 7

Problem 8.17. (a) Let M and N be oriented manifolds, and let ω and η be an n-form and an m-form with compact support, on M and N , respectively. We will orient M ×N by agreeing that v1, . . . , vn, w1, . . . , wm is positively oriented in (M×N)(p,q) ∼= Mp⊕Nq if v1, . . . , vn and w1, . . . , wm are positively oriented in Mp and Nq, respectively. If πi : M ×N →M or N is projection onto the ith f...

Math 225 B : Differential Geometry

Math 225 B : Differential Geometry , Homework 5

Problem 7.8. (a) Let ω ∈ Ω(V ). Show that there is a basis φ1, . . . , φn of V ∗ such that ω = (φ1 ∧ φ2) + · · ·+ (φ2r−1 ∧ φ2r). (b) Show that the r-fold wedge product ω∧· · ·∧ω is non-zero and decomposable, and that the (r + 1)-fold wedge product is 0. Thus r is well-determined; it is called the rank of ω. (c) If ω = ∑ i<j aijψi ∧ ψj, and A is the upper triangular matrix with Aij = aij for i <...

On Furstenberg's Proof of the Infinitude of Primes

1. C. B. Boyer, History of Analytic Geometry, Scripta Mathematica, New York, 1956. 2. J. L. Coolidge, The origin of analytic geometry, Osiris 1 (1936) 231–250, also available at www.jstor. org. 3. M. Ghomi and B. Solomon, Skew loops and quadric surfaces, Comment. Math. Helv. 4 (2002) 767–782. 4. J.-P. Sha and B. Solomon, No skew branes on non-degenerate hyperquadrics, Math. Zeit. 257 (2002) 225...

MATH 281A: Homework #4