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Does somebody have some interesting examples of Courant algebroids which are not exact? By exact I mean one which is of the form $TM\oplus T^\star M$ with the standard bracket twisted by a closed 3-fo …

Hello! Let $E$ and $F$ be two vector bundles and let $f:E\rightarrow F$ be a bundle map. Then the kernel of $f$ is not always a subbundle of $E$. Does somebody have a simple example? Does there exist …

An exact Courant algebroid $E$ is one such that the sequence $0\to T^\star M\xrightarrow{\rho^\star} E^\star\simeq E\xrightarrow{\rho} TM\to 0$ is exact. Here $\rho$ is the anchor of the algebroid. Si …

Hello! Let $E\rightarrow M$ be a holomorphic vector bundle. We denote by $\mathcal{E}$ its sheaf of holomorphic sections and by $\mathcal{O}$ the sheaf of holomorphic functions on $M$. We also denote …

Hello! I am reading an article in which there is the following statement: Let $E\rightarrow X$ be a holomorphic vector bundle. The holomorphic sections of $E$ over a coordinate neighbourhood of $X$ …

Hello! We know that we have an alternative way to define a complex structure on manifold, by means of an integrable almost complex structure. The two points of view are equivalent, this is the conten …

Could somebody provide some simple examples of Courant algebroids coming from Poisson geometry?

A lot of notions in differential geometry have direct meaning in Physics. For example: A Riemannian metric is a way to encode distances on a manifold and in Physics it is the gravitational field. Th …

I would like to add that generalized geometry in the sense of Hitchin is also a good framework for the notion of brane (a very important notion in what physicists call M-theory) and also T-dualities o …

Hello! Let $M$ be an almost complex manifold. Let $TM$ denote its tangent bundle. Then we have the decomposition $TM\otimes\mathbb{C}=T^{1,0}M\oplus T^{0,1}M$ corresponding to the eigenvalues of the …