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I don't know exactly where it appears in the literature, but a coordinate-free interpretation is certainly known, based on the idea of Penrose's twistor construction. The main points can be summarize …

I think that the usual proof goes through in this case, although, obviously, you don't get a diffeomorphism (i.e., $C^\infty$ invertible map) identifying the two volume forms, just a $C^{1+\alpha}$ ma …

Part I: The original question: Now that the question has been clarified, I can answer it. The answer is 'no', there is no CR-isomorphism $\phi: \mathrm{Heis}\to U$ that is unitary on the holomorphi …

But, can't you read this off the table of simple roots for each of the exceptional Lie algebras? For example, $\mathrm{G}_2$ contains $\mathrm{SU}(3)$ as a subgroup and the maximal torus of $\mathrm{ …

Here is my own favorite way of understanding this problem: Let $\bigl(L,[,]\bigr)$ be a $3$-dimensional Lie algebra over a field $K$ (assumed of characteristic $0$ for my comfort, though this probabl …

The question you are asking is a very basic one in the theory of what Élie Cartan called "the method of the moving frame" (in the original French, "la méthode du repère mobile"), so you should be look …

Yes. If $p\in\mathbb{CP}^2$ is a point, you can consider the blowup $X_p$ of $\mathbb{CP}^2$ at $p$ as the space of pairs $(L,q)$ such that $L\subset\mathbb{CP}^2$ is a line passing through $p$ and $ …

This is really just an amplification of Deane's answer. As he points out, it is indeed true that a flat submanifold $M^n\subset \mathbb{R}^{n+1}$ whose second fundamental form is nonvanishing is cano …

An alternative is to simply use Riemann-Roch, which does not depend on the Uniformization Theorem. R-R says that $\ell(D)-\ell(K-D) = \mathrm{deg}(D)-g+1$, where $\ell(D)$ is the dimension of the s …

I think what you are looking for is something like Theorem 9 in the following reference: Ehrenpreis, Leon, A fundamental principle for systems of linear differential equations with constant coefficien …

Probably, you can find a discussion of this in Thurston's notes on hyperbolic 3-manifolds, or maybe some of the expositions by his students. However, what you are asking for is actually pretty simple: …

You might find it useful to make a change of variables to reduce the equation to a more familiar form. For example, if we assume, as we may, that $a$ and $b$ are not equal, then we can substitute $y …

Actually, the only nontrivial case is the Kahler case: If you put on the assumptions you need in order for the Ricci flow to be uniquely defined, then by passing to a cover, you can assume that $(M …

I don't know a reference, but this desired normal form is, indeed, attainable. Here is the argument: Assume given a Kähler form $\omega$ defined on a neighborhood of $0\in\mathbb{C}^n$ and that ther …

I don't have a reference, but there is a conceptual way to see that you only have to check a few constants to get this formula. Notice that it's a statement about a quadratic mapping from the space …