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I can give you a partial answer- a reformulation of your question. Perhaps others will then be able to give a counterexample. (I think it is not true in general.) Let's start from the beginning. A ma …

I believe you can find this in the papers of Lockhart and McOwen. Specifically, check Lockhart, Robert B.; McOwen, Robert C. Elliptic differential operators on noncompact manifolds. Ann. Scuola Norm. …

If $A$ is any section of a vector bundle $E$ over a smooth manifold $M$ and if $\nabla$ is any covariant derivative on $E$, then $\nabla A$ is a section of $T^*M \otimes E$, and this has a natural (po …

Using the additional information that the OP provided in the comments to Yael Fregier's answer, I can elaborate as follows: I still don't know what "special complex manifold" means, but in any case, …

Well, now there is a great textbook for Sasaki-Einstein geometry, by Boyer-Galicki: Sasakian Geometry. Here is a link to the book at Oxford University Press, DOI: 10.1093/acprof:oso/9780198564959.001. …

Just to follow up on Eric's correct answer: when you have an almost complex structure $J$, you can decompose $1$-forms into type $(1,0)$ and $(0,1)$. Locally, you can find a local basis $e^1, \ldots, …

David Speyer already explained it very well. But if you want more details of the explicit calculations, you can look here: http://www.math.uwaterloo.ca/~karigian/teaching/multivariable-calculus/moebi …

This is probably closer in spirit to what you're looking for than what you've received in the comments. If $(V^{2m}, J, \omega, g)$ is Calabi-Yau (which for me means that $J$ is integrable, and the fi …

On a Riemannian manifold, a coordinate system is called "isothermal" if the Riemannian metric in those coordinates is conformal to the Euclidean metric: $$g_{ij} = e^{f} \delta_{ij}$$ My question is …

Let's suppose that $M$ is a connected $1$-dimensional smooth manifold (Haussdorf and paracompact). We know that there are exactly two types, up to diffeomorphism (even up to homeomorphism), namely $\m …

Another excellent book that does a proper job with Noether's Theorem is Peter Olver's Applications of Lie Groups to Differential Equations.

The paper by Apostolov, Calderbank, and Gauduchon that Francesco mentions find different Kaehler structures whose associated Riemannian metrics are conformal to each other. But they correspond to diff …

I would also add the following book: Dominic Joyce, Compact Manifolds with Special Holonomy The early parts of the book include an introduction to the Riemannian geometry of Calabi-Yau manifolds. It …

(I decided to repost multiple comments as an answer. It's partly an answer if I understand the OP correctly.) I think the OP does not necessarily want to conclude that these "local" forms are the res …

[First paragraph has been edited, after Vitali's comments below.] According to one convention, hyperKahler manifolds are not actually quaternionic-Kahler. This is the case if you define hyperKahler as …