# Tagged Questions

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### the push forward of the differential idea of sheaf

This is about the differential idea of a sheaf as a vector bundle $E$ (not necessarily locally trivial) on a real manifold $M$ with a zero curvature connection $\nabla$. The zeroth cohomology is then ...

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46 views

### Construction of a Hermitian metric on a locally free subsheaf which is not a subbundle

Assume $X$ is a smooth projective curve over $\mathbb{C}$ and let $\mathcal{E}$ be a locally free sheaf of rank 2 on $X$. We pick a closed point $x\in X$ and a surjection $f: \mathcal{E}\rightarrow ...

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**1**answer

184 views

### Exercises around Diffeological Spaces or a Diffeologic Atlas Theory

Is there a Book or a bunch of exercises to get used to diffeological spaces from a practical point of view? It seems to me that papers on this topic are mostly concerned with their very good ...

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**1**answer

298 views

### Is this Sequences of Complexes of Sheaves Exact?

So in another question of mine there is a sequence of complexes of sheaves which the author asserts is exact.
Let $K^{\bullet} = \underline{\mathbb{C}}^* \ \underrightarrow{d\ log} \ ...

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**5**answers

3k views

### Is there a sheaf theoretical characterization of a differentiable manifold?

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 ...

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478 views

### What essential property justifies the name “derivative”?

Most, if not all, of the notions of derivative that I have so far seen have the property that they are locally defined -- meaning that the derivative of a map-type object at a point depends on the map ...

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**4**answers

1k views

### The tangent bundle to an infinite-dimensional manifold

Suppose that $A,B$ are smooth ($\mathrm C^\infty$) manifolds, and denote by $\hom(A,B)$ the set of $\mathrm C^\infty$-maps $A \to B$. It is a perfectly well-defined set, but often one wants more. ...

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258 views

### Is a submodule of the sheaf of sections of a smooth vector bundle necessarily finitely generated?

Let $X$ be a finite-dimensional smooth manifold, $\mathcal C^\infty(X)$ its algebra of smooth functions, $V\to X$ a finite-dimensional smooth vector bundle, and $\Gamma(V)$ the space of smooth ...

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**4**answers

705 views

### Is there an easy way to describe the sheaf of smooth functions on a product manifold?

A smooth structure on a manifold $M$ can be given in the form of a sheaf of functions $\mathcal{F}$ such that there is an open cover $\mathcal{U}$ of $M$ with every $U\in \mathcal{U}$ isomorphic ...

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1k views

### De Rham decomposition theorem, generalisations and good references

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 ...

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964 views

### Loop Spaces as Generalized Smooth spaces or as Infinite dimensional Manifolds?

There are two ways to define smooth mapping spaces and I want to know how they compare?
Let's take the concrete special case of free loops spaces. I think this is the most studied example so will ...

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1k views

### What is a section?

This question comes out of the answers to Ho Chung Siu's question about vector bundles. Based on my reading, it seems that the definition of the term "section" went through several phases of ...