Doubts on differential geometry from Lang's book

Question

I am an undergraduate self-studying Differential Geometry from Lang's book "Fundamentals of Differential Geometry". I think there will be a series of doubts from my side, starting with this one.

In the chapter on MANIFOLDS, Lang has said a lot of things which I do not understand why are true.

Background: Consider $X$ to be a set and $(U_i,\phi_i)$ to be a chart, where $U_i\subset X$ and $\phi_i:U_i\to E_i$ where $E_i$ is a Banach space, and $\phi_i$ is a bijection to an open set $\phi_i(U_i)$ of $E_i$.

The statements that I do not understand:

If two charts $(U_i,\phi_i)$ and $(U_j,\phi_j)$ are such that $U_i\cap U_j$ is non-null, then by considering the derivative of $\phi_j\phi_i^{-1}:\phi_i(U_i\cap U_j)\to \phi_j(U_i\cap U_j)$, we see that $E_i$ and $E_j$ are toplinearly isomorphic i.e. there exists a continuous linear isomorphism between $E_i$ and $E_j$.

Why is this true? I know that $\phi_j\phi_i^{-1}$ is an isomorphism from $\phi_i(U_i\cap U_j)$ to $\phi_j(U_i\cap U_j)$. Why then will the two Banach spaces $E_i$ and $E_j$ be top-linearly isomorphic?

The set of points $x\in X$ such that there exists a chart $(U_i,\phi_i)$ at $x$ such that $E_i$ is toplinearly isomorphic to a given space $E$ is both open and closed.

Why is this true?

Therefore, on each connected component of $X$, we can assume that we have an $E-$atlas i.e. collection of charts, for some fixed $E$.

Why is this true, then? I think this has more to do with topology than differential geometry, but want to clarify.

Answer 1

1) The derivative is a continuous linear map between tangent spaces. Because the map is a diffeomorphism, its derivative must be an isomorphism. The tangent space to any point of a Banach space is the Banach space itself.

2) Clearly it's open. If $x_n \to x$, and there is an $E$-chart around each $x_n$, then pick an $F$-chart around $x$. Necessarily this contains the $x_n$ for large $n$. Thus the tangent space at $x_n$ is isomorphic both to $E$ and $F$, and so $E \simeq F$ as desired.

3) A set that is both open and closed is a union of connected components.

Answer 2

As an undergraduate studying differential geometry I would highly suggest that you pick a book by a geometer or someone close to the field. As I understand, Lang was a number-theorist and wrote most of these textbooks as a challenge to himself, scratching the surface of each topic outside of his field. Some books I find useful are:

(1) Barnett and O'neill Differential Geometry (excuse my misspelling)

(2) Loring Tu Manifolds

(3) John Lee Introduction to Smooth Manifolds