Search Results

On a smooth manifold $M$ one can define various algebraic structures, natural with respect to diffeomorphisms: the differential graded-commutative algebra $\Omega(M)$ of differential forms on $M$; t …

The constant rank theorem says that if $f\colon M→N$ is a smooth map whose rank equals some fixed $k≥0$ at any point of $M$, then, locally with respect to $M$ and $N$, the map $f$ assumes the easiest …

(1) This is a highly productive way of looking at smooth manifolds. It is responsible for synthetic differential geometry and derived smooth manifolds. Both of these subjects heavily rely on this iden …

Another application of stacks is in synthetic differential geometry. Start with the opposite category of germ-determined finitely generated C^∞-rings and equip it with the appropriately defined Groth …

What is being described in the main post is simply the de Rham (crossed) complex valued in a Lie group (not necessarily commutative). See, for example, Section 6.2 in Anders Kock's Synthetic Geometry …

Smooth manifolds of negative dimension are defined in derived geometry. Recall that if A→M and B→M are two transversal submanifolds of codimension a and b respectively, then their intersection C is ag …

Stacks are used in complex analysis, for example. See the papers by Finnur Lárusson, in particular, Excision for simplicial sheaves on the Stein site and Gromov's Oka principle, which shows that havi …

A spin structure on a real vector space V equipped with a real quadratic form μ is an invertible bimodule (i.e., a Morita equivalence) from Cl(V,μ) to Cl(Rdim(V),ν). Here ν is the direct sum of dim(V) …

Your definition of a smooth manifold still uses atlases in a slightly disguised way because it amounts to saying that a smooth manifold is a topological manifold with an open cover whose elements are …

The category of line bundles (possibly with connection) on a smooth manifold M can be defined in two different ways: The first definition uses transition functions that satisfy a cocycle condition (po …

A folk theorem says that star-shaped open subsets of R^n are diffeomorphic to R^n. Is there a citeable reference for a proof of this result? For the sake of being definite, let's say that “citeable” m …

The fiberwise Stokes theorem says that given a differential form on a smooth fiber bundle whose fibers have boundary, the difference between the fiberwise integral of the differential and the differen …

every group can be declared a Lie group if one allows the more general definition since one is permitted to have an uncountable number of connected components. A manifold is paracompact if and o …

Apart from Nestruev's book which is good but unfortunately very elementary, I recommend you to take a look at Ramanan's Global Calculus. Ramanan almost manages to avoid coordinates except for a few pl …

“Diffeology” by Patrick Iglesias-Zemmour is probably the closest match. He develops differential forms and de Rham cohomology, fiber bundles, connections, and symplectic geometry in the language of di …