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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
comment What does it mean for a subset of $L^{\infty}(G)$ to separate the points of $G$?
Did you see the term "separate points" in the context of $M \subset L^\infty(G)$ somewhere in particular, or are you trying to invent a new notion?
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?
Apr
14
comment What does a skew second fundamental form geometrically mean?
If you isometrically embed a Riemannian 2-manifold into a higher dimensional Riemannian manifold, then the second fundamental form, $\text{II}$, will be a symmetric bilinear form. If $\text{II}$ were also skew-symmetric, then it would be zero, meaning that the surface is "totally geodesic." The "totally geodesic" property has a few geometric interpretations. That said, I don't know what you mean by a "non-Riemannian embedding space."
Apr
14
comment What does a skew second fundamental form geometrically mean?
What exactly do you mean by a non-Riemannian embedding space? And by "skew," do you mean that you want the second fundamental form to be skew-symmetric?
Apr
11
comment Differential Form Pullback Definition
@mathfinalshelp: Yes, that's right.
Apr
10
awarded  Nice Question