Vector fields, flows, and the Lie derivative

Let M be a smooth real manifold. A smooth vector field V on M can be considered as a function from C^{\infty}(M) to C^{\infty}(M). Every function f : M \to \mathbb{R} at every point p \in M is by a vector field (which implicitly associates a tangent vector at every point) taken to some real value, which one can think of as the directional derivative of f along the tangent vector. Moreover, this varies smoothly with p.

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Sheaves of holomorphic functions

I can sense vaguely that the sheaf is a central definition in the (superficially) horrendously abstract language of modern mathematics. There really does seem to be quite a distance, between crudely speaking, pre-1950 math and post-1950 math in the mainstream in terms of the level of abstraction typically employed. It is my hope that I will eventually accustom myself to the latter instead of viewing it as a very much alien language. It is difficult though, and  there are in fact definitions which take quite me a while to grasp (by this, I mean be able to visualize it so clearly that feel like I won’t ever forget it), which is expected given how long it has taken historically to condense to certain definitions golden in hindsight. In the hope of a step forward in my goal to understand sheaves, I’ll write up the associated definitions in this post.

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Urysohn metrization theorem

The Urysohn metrization theorem gives conditions which guarantee that a topological space is metrizable. A topological space (X, \mathcal{T}) is metrizable is there is a metric that induces a topology that is equivalent to the topological space itself. These conditions are that the space is regular and second-countable. Regular means that any combination of closed subset and point not in it is separable, and second-countable means there is a countable basis.

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Path lifting lemma and fundamental group of circle

I’ve been reading some algebraic topology lately. It is horrendously abstract, at least for me at my current stage. Nonetheless, I’ve managed to make a little progress. On that, I’ll say that the path lifting lemma, a beautiful fundamental result in the field, makes more sense to me now at the formal level, where as perceived by me right now, the difficulty lies largely in the formalisms.

Path lifting lemma:    Let p : \tilde{X} \to X be a covering projection and \gamma : [0,1] \to X be a path such that for some x_0 \in X and \tilde{x} \in \tilde{X}, Continue reading “Path lifting lemma and fundamental group of circle”

Another characterization of compactness

The canonical definition of compactness of a topological space X is every open cover has finite sub-cover. We can via contraposition translate this to every family of open sets with no finite subfamily that covers X is not a cover. Not a cover via de Morgan’s laws can be characterized equivalently as has complements (which are all closed sets) which have finite intersection. The product is:

A topological space is compact iff for every family of closed sets with the finite intersection property, the intersection of that family is non-empty.

Grassmannian manifold

We all know of real projective space \mathbb{R}P^n. It is in fact a special space of the Grassmannian manifold, which denoted G_{k,n}(\mathbb{R}), is the set of k-dimensional subspaces of \mathbb{R}^n. Such can be represented via the ranges of the k \times n matrices of rank k, k \leq n. On application of that operator we can apply any g \in GL(k, \mathbb{R}) and the range will stay the same. Partitioning by range, we introduce the equivalence relation \sim by \bar{A} \sim A if there exists g \in GL(k, \mathbb{R}) such that \bar{A} = gA. This Grassmannian can be identified with M_{k,n}(\mathbb{R}) / GL(k, \mathbb{R}).

Now we find the charts of it. There must be a minor k \times k with nonzero determinant. We can assume without loss of generality (as swapping columns changes not the range) that the first minor made of the first k columns is one of such, for the convenience of writing A = (A_1, \tilde{A_1}), where the \tilde{A_1} is k \times (n-k). We get

A_1^{-1}A = (I_k, A_1^{-1}\tilde{A_1}).

Thus the degrees of freedom are given by the k \times (n-k) matrix on the right, so k(n-k). If that submatrix is not the same between two full matrices reduced via inverting by minor, they cannot be the same as an application of any non identity element in GL(k, \mathbb{R}) would alter the identity matrix on the left.

I’ll leave it to the reader to run this on the real projective case, where k = 1, n = n+1.