Math 225 B : Differential Geometry

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...

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 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 <...

Notes for Math 535a: Differential Geometry