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revised What are some simple examples illustrating the definition of “cover”
edited body
May
22
comment Good sources to learn about Geometric Analysis
@AbrahamRabinowitz: To be honest, I wish I knew. There's a pretty substantial chapter on it in Evans' book on PDE, but other than that I don't know.
May
21
answered Good sources to learn about Geometric Analysis
May
20
comment Why are diffeomorphism-invariant PDE not elliptic?
@aes: Thank you. Could you please turn your comment into an answer?
May
12
asked Why are diffeomorphism-invariant PDE not elliptic?
Apr
26
comment General setting of Varadhan's result for distance functions and heat kernels
Interesting! Looking forward to answers. If you end up not getting an answer, though, you might consider asking this question on MathOverflow.
Apr
22
answered symplectic manifolds
Apr
22
comment symplectic manifolds
Whether there exist non-degenerate $2$-forms (not necessarily closed) is a purely topological question; a complete characterization of such manifolds can be given in terms of characteristic classes. But it's not clear which of those also support a non-degenerate $2$-form which solves $d\omega = 0$. (Just as it is not clear, for example, which smooth manifolds admit integrable complex structures.)
Apr
22
comment symplectic manifolds
From a more analytic perspective, you're asking for a global solution $\omega \in \Omega^2(M)$ to the differential equation $d\omega = 0$ on $M^{2n}$ which also satisfies $\omega^n \neq 0$. In general, I don't think it's easy to know when a differential equation has global solutions or not.
Apr
20
comment Lipschitz implies bounded derivative?
To be clear, we're assuming that $f$ has codomain $\mathbb{R}$, then? I ask because the notation $d_y$ suggests a more general context.
Apr
19
comment The even-numbered coefficients of the Maclaurin series of $ \frac{1}{\cos(x)} $ are odd integers.
The OP seems to be asking why the non-zero coefficients (associated to the even powers) are odd integers.
Apr
19
comment Open Unit Ball diffeomorphic to the Open Unit Cube
Just an aside, but the set $B$ is usually called the "open unit ball," and almost never called the "open unit sphere".
Apr
19
comment A differentiable manifold of class $\mathcal{C}^{r}$ tangent to $E^{\pm}$ and representable as graph
To be clear, part of your question is that you don't know what a "differentiable manifold of class $C^r$" is, yes?
Apr
18
comment What does it mean for partial derivative to be continuous and how does that imply differentiability?
The question in the title is different from the question in the post. Are you asking (1) why the continuity of partial derivatives implies differentiability (which is true), or (2) why differentiability implies the continuity of partial derivatives (which is false)?
Apr
14
comment Key differences between almost complex manifolds and complex manifolds
@Bilateral: Yes, that's exactly right, thanks. Sorry, I was being careless.
Apr
14
awarded  Revival
Apr
14
comment Möbius strip as a non-trivial principal bundle
Are you regarding the Mobius strip as a principal $G$-bundle for some Lie group $G$ (with fibers being copies of $G$), or as a vector bundle (with fibers being vector spaces)?
Apr
14
answered Key differences between almost complex manifolds and complex manifolds
Apr
14
revised How much does Proof writing improve over the years?
added 26 characters in body
Apr
14
answered How much does Proof writing improve over the years?