First steps with Vector Bundles

Question

I've concluded a differential geometry course (mainly covering classical results about parametric surfaces or diff. surfaces in $\mathbb{R}^3$, for example Gauss' Theorema Egregium or geodetics) which ended with a two weeks digression over "abstract differential geometry" (basic definitions, one rapid view over some basic results, local immersion/submersion theorem and two words about vector bundles and fields...). I wanted to view something more about vector bundles and i started reading the second volume of Hatcher "Vector Bundles and K-Theory" (mainly because I liked the way Hatcher explains the ideas in the few chapter I read in his first book). But I've some problem in understanding his introduction in chapter 1. We are speaking about $TS^2$, the set of tangent spaces of $S^2$ (the sphere in $\mathbb{R}^3$). After equipping it with a topology (the topology of a subspace of $S^2 \times \mathbb{R}^3$) he says:

One can think of $TS^2$ as a continuous family of vector spaces $P_x$ for $x \in S^2$ parametrized by points of $S^2$.

My problem is the word "continuous", maybe is just the fact this is the first time I'm studying this kind of objects "standing alone" but I don't know how to prove this continuity or how to visualizing it.

I hope this is not a silly question, and if someone has some advices (for example, some books who treat this topic with a lower level or so on) they are greatly appreciated

Answer 1

The differential structure of a smooth manifold $M$ is given by charts $\varphi: U\to\mathbb R^n$ on open sets $U\subseteq M$. These charts basically allow us to flatten $M$ locally. Thus, when we want a topology on $TM$, we want $TU\subseteq TM$ to be homeorphic to the tangent bundle of the flattened $\varphi(U)$ $$ TU \cong T\varphi(U) \cong \varphi(U)\times \mathbb R^n, $$ where the tangent spaces of $\varphi(U)$ are glued together in the obvious way (since $\varphi(U)$ is just an open set in $\mathbb R^n$). Since $M$ is covered by charts and we know the topology of $TM$ over each chart, we get a topology on $TM$ by gluing the $TU$ together, again in the obvious way: $$ TM = \left(\bigsqcup TU\right)\big/{\sim} $$ where $\sim$ identifies the tangent spaces in $U\cap U'$ for overlapping charts.

Does this help to understand in what sense this family of tangent spaces is continuous?

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Update, Another way of looking at this "continuity" in Hatchers model:

In the context of $TS^2$ as a subspace of $S^2\times\mathbb R^3$ like Hatcher introduces it, you have a tangent plane in $\mathbb R^3$ for every $x\in S^2$. If you change $x$ only by a small amount, the corresponding tangent plane will also just move by a small amount. Another way to phrase it: When $y$ approaches $x$ in $S^2$, the tangent planes at $y$ come closer and closer to tangent space at $x$.

Answer 2

There is nothing to "prove" about the continuity of the vector spaces involved in a vector bundle. This is essentially the content of the definition of the term "vector bundle." I believe this sentence is just an intuition building reinforcement from the author. Constructively speaking, one begins by wanting to capture the idea of vector spaces varying continuously over a space, thinks for a while, and (ideally) writes down the textbook definition that is currently used.

Answer 3

This is just an informal quick reformulation of the formal definition, found for example in chap. 2 of Milnor+Stashef: If $X$ is a topological space, a vector bundle over $X$, say real of rank $n$, is a toplogical space $E$ and a map $p:E\to X$ such that (1) $E_x=p^{-1}(x)$ is a real vector space of dimension $n$; (2) (here comes the "continuous family of vector spaces") for every $x\in X$ there is an open set $U\subset X$ containing $x$ and a homeomorphism $p^{-1}(U)\to U\times \mathbb R^n$ whose restriction to each $E_y$, $y\in U$, is a linear isomorphism unto $\{y\}\times\mathbb R^n$.

The standard first non-trivial example is the tangent bundle of a differentiable manifold, where the homeomorphisms of property (2) above (called "local trivializations" of $E$) are provided by the derivatives wrt local coordinates. This is an easy exercise.