Today was characterized by two words: hungry and sleepy. It’s funny that whenever I engage in a modest amount of physical exercise, my body gets back at me with ten times the magnitude. I guess that’s the crucial difference between Asians and other obesity-prone races: we are leaner. Anyhow, I am increasingly disappointed by the food quality in the dining hall. Maybe it should only be used sparingly. The asian chopstix place is still the best and most conforming to my stomach.
So back to math, the major achievement of this morning was a direct attack at the Riemannian topology problem I mentioned yesterday. I broke it into three cases, according to the type of intersection of the plane with the tangent plane of the manifold at a point p, as embedded in the ambient space R^n. The case when the intersection is 0 is easy, because we can conclude that in a neighbor the intersection of the plane with the manifold M must consist of just one point. When the exogenously given plane is contained in the tangent plane of the manifold TpM, we premsumably also get locally one point intersection, by analogy with the tangent plane of a sphere, which is the quintessential example of a manifold with positive sectional curvature. The only remaining nasty case is when the intersection is not 0 and yet the plane is not contained in the tangent space. Currently I have no clue how to deal with it; the example with a plane intersection a sphere transversally in R^3 seems to suggest how complicated this situation could be. And the first nontrivial example needs some 4 dimensional visualization skill. So one day when I am hard pressed, I might email William Thurston. Also I realized that O(n) might be easily proven to be positively curved (which would then imply curvature bounded below by a positive constant by compactness), because of the fact it’s contained in S^(n^2-1)(sqrt(n)), hence the geodesic curves along two independent directions of the tangent plane will lie on one side of the tangent plane, which implies nonnegative curvature. And together with the fact that they can’t be lines, we get positivity.
I was completely stagnant shall we say, in deciphering Louis Arguin’s paper on robustly quasistationary random overlap structures. So I emailed Arguin with apology and gratitude.
Alright, enough for today.