All Questions

I'm going through the crisis of being unhappy with the textbook definition of a differentiable manifold. I'm wondering whether there is a sheaf-theoretic approach which will make me happier. In a ...

De Rham decomposition theorem states that every simply-connected Riemannian manifold $M$ that admits complementary sub-bundles $T'(M)$ and $T''(M)$ of its tangent bundle parallel with respect to the ...

I fairly understand the fiber bundles, both the mathematical concept of fiber bundles and the physics use of fiber bundles. Because the fiber bundles are tightly connected to the gauge field theory in ...

Given a vector bundle $E \to M$ with connection $\nabla$, we get a twisted de Rham sequence using the exterior covariant derivative: $$0 \to \mathcal{E} \xrightarrow{d^\nabla} \Omega^1_M \otimes \...

I'm reading bits and pieces of Kolar, Michor, & Slovak's Natural Operations in differential Geometry, and I'm having "doubt" about some of the definitions. All I'm trying to do is sheafify some of ...

It is known that the natural functor of smooth functions from the category of smooth manifolds into the category of locally ringed spaces is a full embedding (see, for example, here). Is a similar ...