Complex Analytic and Differential Geometry 2024, 45: Differential Calculus on Manifolds

I didn't think that last book was too painful to read, but perhaps that is because it did rely very heavily on Topology and Group-Theory concepts, rather than what I suspect is differential geometry. On a second attempt at finding a book, I stumbled across this: Complex Analytic and Differential Geometry which sounds much more like something I'll have a hard time with. It starts comfortable in the realm of analysis, but since it's been a while since I've done this specific type, I'll start from the very beginning for this one. Luckily, we can just borrow a lot of the language from the previous book.

A differentiable manifold M with dimension m, of class Cᵏ is a topological space with an atlas of class Cᵏ and values in ℝᵐ. An atlas is a family of homeomorphic differential charts τ with

whose components are local coordinates on U. Cʳ(Ω⊂M, ℝ) for an open set Ω is a set of functions f of Cʳ so that the compositions with a differential chart is of class Cʳ as well. Cʳ(Ω, ℝ) has a Cʳ extension to some neighborhood of Ω. A manifold is orientable iff there is an atlas whose transition maps all preserve orientation. A continuous form u of maximum degree m = dim[M] with compact support in an open set,

which is independent of the choice of coordinates.

Let u be a differential form on [0, 1] ⨉ M with operator K

Two smoothly homotopic infinitely differentiable maps F, G

On contractible manifolds, smooth homotopies of order 0 are the real space, and smooth homotopies of order p are exactly 0 for p greater or equal to 1.

On a manifold, the exterior derivative for a current on a manifold with respect to some T in the subspace D of elements with topology defined by all seminorms p with support contained in M and the resulting induced topology, follows

The definition of the wedge product follows from the exterior derivative.

Two orientable manifolds M, N with dimensions m, n and an infinitely differentiable map F from M to N defines a pull-back morphism. For compact support of T on its intersection with the inverse of F, then there is a unique "direct image" F*T

For every T with proper support on F, the direct image of F*T on N the following four statements are true

For submersions F from M to N, a consquence of the continuity statement, there is always an inverse image of the form F*T