14

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 Grothendieck topology, then pass to ∞-stacks. The resulting category (known as the Dubuc topos) contains all smooth manifolds, is a Grothendieck ∞-topos (so in ...


11

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 having the Oka–Grauert property for a complex manifold X is equivalent to the condition that the presheaf of spaces of holomorphic maps into X is an ∞-stack in the ...


7

A few years ago, Bernstein wrote a note with a new approach to representation theory of algebraic groups using the langage of stacks.


7

A few remarks: First, in a sense, (special cases of) this (are) is very commonly studied. Because a bilinear differential form $g$ as you have defined it can naturally be written as a sum $g = \sigma + \alpha$ where $\sigma$ is symmetric and $\alpha$ is skew-symmetric, you are, equivalently, asking about the geometry of the pair $(\sigma,\alpha)$. The ...


6

Are there any places one has to be careful to not allow large categories? No. For the purposes of forming the 2-category of algebraic/topological/differentiable stacks, or more generally, some kind of presentable stacks over a large category there are no size issues. Naively, the 2-category of stacks on $S$ is carved out from the presheaf category $[S^{op},\...


5

Stacks over the category locales are very interesting for topos theory: A big success of topos theory is the fact that the $(2,1)$-categories of Grothendieck toposes and geometric morphisms between them embedded as a reflective full subcategory of the category of localic stacks, that is stack on the category of locales. It is in fact a full subcategory of ...


5

There are two notions of stack. The one you mention is a sheaf of groupoids. Sometimes these come up on their own. The other notion is a geometric object, often a "bad quotient." This object can be represented as a sheaf of groupoids, but that is only a technical tool. If you had other tools, you might use them instead. For example, if you had a foliation of ...


4

As noted in the comments, by the spherical symmetry, the distribution of the dot product in both parts of your question is the same that of $X\cdot(1,0,\dots,0)$. Moreover, the distribution of $X$ is the same as that of the random vector $$\frac{Z}{\sqrt{Z_1^2+\dots+Z_d^2}},$$ where $Z=(Z_1,\dots,Z_d)$ is a standard normal random vector. So, the ...


4

In terms of the Hamiltonian $H$ the antisymmetry follows simply: Majorana fields $\Psi(x,t)$ are real, and they satisfy a real wave equation $$\partial\Psi/\partial t =-iH\Psi,$$ where $iH$ is real, hence $H$ is imaginary. Since $H$ must also be Hermitian, it means that $H^T=-H$ (antisymmetric). More explicitly, $H=\gamma_{\rm M}^\mu \partial_\mu$, with $\...


3

There is a vast literature on singularities, like: Gibson, Christopher G.; Wirthmüller, Klaus; du Plessis, Andrew A.; Looijenga, Eduard J. N. Topological stability of smooth mappings. Lecture Notes in Mathematics, Vol. 552. Springer-Verlag, Berlin-New York, 1976 Arnold, V. I.; Gusein-Zade, S. M.; Varchenko, A. N. Singularities of differentiable maps. Volume ...


3

There are some applications of Gauß-Bonnet in foliations and laminations of 3-manifolds. The first result that comes to mind is Candel’s Uniformization Theorem, which gives necessary and sufficient conditions for a lamination to admit a leafwise hyperbolic metric. The proof uses Gauß-Bonnet in a nontrivial way. (A shorter exposition is in Chapter 7 of ...


3

Let $C_\alpha$ denote the $n$-dimensional closed Euclidean solid cone with the cone angle $\alpha\in (0, \pi)$. Let $X_\alpha$ be the metric space obtained by gluing two copies $C^\pm_\alpha$ of $C_\alpha$ at their tips, and equipped with the natural path-metric. Let $o\in X_\alpha$ denote the common tip of the cones. I will use it as the center of the ...


2

The first bullet is definitely explained in the book! Surely around where it was introduced, it has to do with summing over all spin-c structures. We need to pass to the completion because the 4-manifold can have infinitely many spin-c structures that would need to be used. The second bullet, yes. In general we should not expect results concerning ...


2

The answer in Modification of Morse lemma with two functions shows that this doesn't work in general (depending on what you mean)


2

This is more of a long comment. As far as I know, the first question is a very difficult (and interesting) one. Here are a few examples of diffusions in $\mathbb R^3$ that are not diffeomorphic in your sense, and may give an idea of the difficulties ahead: the standard Brownian motion in $\mathbb R^3$; the Brownian motion along planes $z=c$, i.e. $X_t=(B^...


1

There are "bundle gerbes" (introduced by Murray), which are a particular kind of stacks. People study connections on them, generalizing connections on principal bundles.


1

I like Chern's short and easy Vector bundles with a connection, in Chern, Global Differential Geometry, M.A.A., 1989. Chern focuses on vector bundles but does so by using simple examples of principal bundles. Chern gives some serious theorems that motivate learning the subject.


1

This is not a complete answer to your question. However, there is one fairly general case in which $h$ will vanish, which is when $(M,g, \nabla)$ is a statistical manifold. A statistical manifold is a Riemannian manifold $(M,g)$ with an affine connection $\nabla$ satisfying $$ (\nabla_X g)(Y,Z) =(\nabla_Y g)(X,Z) $$ for any vector fields $X,Y$ and $Z$. ...


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