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A coordinate-free proof of the inverse function theorem in the finite-dimensional case is provided by Theorem 19.6 in "Topological Geometry" by Ian R. Porteous. In general, the cited book is an expos …

How can we characterize the algebras (at least within all the C^∞(M)'s), that come from compact manifolds? An algebra of the form C^∞(M) corresponds to a compact manifold if and only if all of it …

The functor from the category of smooth manifolds to to the category of real algebras that sends a manifold M to C^∞(M) is fully faithful, hence it is an equivalence of categories of smooth manifolds …

This is true for infinitely differentiable curves: if a map sends smooth curves to smooth curves, then it is smooth, by a theorem of Boman from 1967: Jan Boman. Differentiability of a Function and of …

The étale space construction produces non-Hausdorff and nonparacompact spaces (e.g., smooth manifolds) in many practical examples that have nothing to do with algebraic geometry. The étale space is …

We can give a complete classification of (candidates for) delta-distributions on a smooth manifold $M$ at point $p$. Specifically, a delta-distribution is a smooth linear functional on the space of sm …

The proof of this fact is available in modern textbooks. For example, see Theorem 7.16 in Jet Nestruev's Smooth Manifolds and Observables (Second Edition, 2020). In fact, the cited book contains a lot …

An atlas is a sheaf on the site of cartesian spaces (the site with objects R ), such that ... One can certainly define smooth manifolds in such terms. The cartesian site has finite-dimensional …

As far as I am aware, there is nothing in the literature that treats Riemannian or pseudo-Riemannian metrics on derived smooth manifolds. However, there is an extensive treatment of symplectic structu …

There are many such results. Consider some smooth manifolds M and N. The internal hom Hom(M,N) is a sheaf on smooth manifolds. We can compute its tangent bundle, and it turns out that the tangent spac …

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 …

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 …

Suppose $d≥0$, $m≥0$, $n≥0$, and $\def\R{{\bf R}} f\colon \R^m→\R^n$ is a smooth map whose rank at any point of $\R^m$ is at most $d$. Here and below, smooth means infinitely differentiable. Can we fi …

Recall that a Weil algebra is a finite-dimensional real unital algebra that admits exactly one homomorphism to R. Such algebras form the basis of the Weil approach to differential geometry, pioneered …

For the purposes of this question, let's say that a blurring of a smooth triangulation $T$ of a smooth manifold $X$ is a smooth homotopy $h\colon [0,1] \times X \to X$ such that $h_0=\operatorname{id} …