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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 …

Yes. A beautiful conceptual coordinate-free proof is presented by Berwick-Evans in https://arxiv.org/abs/1310.5383. It obtains both sides of the Chern–Gauss–Bonnet theorem as two limits of a partitio …

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 …

Would this be of any interest to solve some geometric questions. Is there a notion of "derived stack" in the differential geometry setting. The notion of a derived stack in the setting of differenti …

After integration we have a number so isn't it a function? Differential $k$-forms can be integrated along a submersion with $d$-dimensional fibers, which yields a differential $(k-d)$-form. Fiberwis …

I will answer the new version of the question: Does $\ker(\varphi_{*,a})=\ker (\phi_{*,a})$ for all $a\in P$, where $\phi_{*,a}:T_aP\to T_{\phi(a)}X_0$ and $\varphi_{*,a}:T_aP\to T_{\varphi(a)}Y_0$ a …

For compact orientable manifolds, this is accomplished by the notion of a spectral triple. See the question Commutative spectral triples for additional information, including a precise statement of th …

For example: is the notion of a smooth envelope of a geometric R-algebra F well-defined if F lacks a unit? Yes. Recall the construction: $F$ is geometric if the Gelfand homomorphism $$\def\Map{\mat …

Smooth manifolds are affine, thus the sheaf of smooth functions is determined by its global sections. Now C^∞(M×N)=C^∞(M)⊗C^∞(N). The tensor product here is the projective tensor product of complete l …

Substituting $f=f_1f_2$ in the definition of a derivative endomorphism immediately implies that $D_M$ is a derivation, using the fact that $g_1ψ=g_2ψ$ for all vector fields $ψ$ implies $g_1=g_2$, wher …

Yes, Lp-spaces can be defined for arbitrary hermitian vector bundles. For the sake of convenience I denote Lp=L1/p (see this answer for a motivation), in particular L0=L∞ and L1/2=L2. As explained in …

Yes. If $E→M$ and $F→M$ are vector bundles over a smooth manifold $M$, then differential operators $E→F$ of order less than $k≥0$ can be identified with sections of a finite-dimensional vector bundle …

The morphism $$\def\T{{\rm T}} φ:S→\T X$$ can be identified with the homomorphism of algebras $$\def\Ci{{\rm C}^∞} \Ci(\T X)→\Ci(S).$$ The algebra $\Ci(\T X)$ can be identified with the $\Ci$-symmetri …

(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 …

See Theorem 1 in Anders Kock's paper “Differential forms as infinitesimal cochains”, which is devoted precisely to this question. Specifically, the map b in the formula (1) establishes an explicit bij …