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Let $P(M,G)$ be a principal bundle. Let $V$ be a finite dimensional vector space (over $\mathbb{R}$). Let $\rho:G\rightarrow GL(V)$ be a representation. A $V$-valued differential $r$-form $\varph …

The following is an excerpt from Marco Gualtieri's thesis A central theme of this thesis is that classical geometrical structures which appear, at first glance, to be completely different in nature, …

This is from the paper Representations up to homotopy of Lie algebroids by Camilo Arias Abad and Marius Crainic Let $M$ be a smooth manifold. Let $E\rightarrow M$ be a vector bundle. A connection on …

It turns out there is a well defined notion of integrability of a G-structure on a manifold. Thanks to the user Thomas Rot who has given the reference Linear $G$-structures by examples Definition $2.1 …

In the lecture notes A brief introduction to Dirac manifolds, Henrique Bursztyn recall the notion of a Courant bracket on section of the generalised tangent bundle $TM\oplus T^*M$. For $X+\xi, Y+\eta\ …

I am learning about "derived manifolds" from talks of Ping Xu in Higher Structures in Geometry and Mathematical Physics. As far as I understand, one of the main philosophy behind the introduction of d …

I am trying to read read the paper Courant Algebroids and Strongly Homotopy Lie Algebras by Dmitry Roytenberg, Alan Weinstein. Let $M$ be a smooth manifold. A Courant algebroid on $M$ is a vector bund …

I am trying to see some standard examples of Lie $2$-algebroids. The first entry in Google search takes me to Madeleine Jotz Lean's work Lie 2-algebroids and matched pairs of 2-representations — a geo …

Let $(\Gamma_1\rightrightarrows \Gamma_0)$ be a Lie groupoid. There are many places which define the notion of connection on a Lie groupoid. As far as I have seen, there is no mention of these not …

Vague question is the following: Is there a classifcation of Lie groupoids? Slightly less vague question is the following: Is there a (short?) list of "types" of Lie groupoids such that any arbitra …

This is about the notion of representations upto homotopy of Lie algebroids. I am following the reference Representations up to homotopy of Lie algebroids by Camilo Arias Abad and Marius Crainic. Let …

I understood gerbes as generalization of line bundle here. In this, I am trying to understand how to generalize notion of trivialization of line bundle to the notion of trivialization of gerbes. I am …

Given a manifold $M$, the collection of all automorphisms of $M$, denoted by $\text{Aut}(M)$ forms a Lie group. Do we have similar setting in case of Lie groupoid? Is there "a structure" whose "auto …

A Poisson structure on a smooth manifold $M$ is a map $C^\infty(M)\times C^\infty(M)\times C^\infty(M)$ satisfying certain conditions. For a vector space $V$, we can talk about a Poisson structure on …

It turns out that linear Poisson structure need not (or should not) give a Poisson structure on the fibers. But, the conclusion that a linear Poisson structure on a vector bundle $E\rightarrow M$ shou …