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 Apr26 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. Apr22 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? Apr22 answered symplectic manifolds Apr22 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.) Apr22 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. Apr20 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. Apr19 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. Apr19 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". Apr19 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? Apr18 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)? Apr14 comment Key differences between almost complex manifolds and complex manifolds @Bilateral: Yes, that's exactly right, thanks. Sorry, I was being careless. Apr14 awarded Revival Apr14 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)? Apr14 answered Key differences between almost complex manifolds and complex manifolds Apr14 revised How much does Proof writing improve over the years? added 26 characters in body Apr14 answered How much does Proof writing improve over the years? Apr14 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." Apr14 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? Apr11 comment Differential Form Pullback Definition @mathfinalshelp: Yes, that's right. Apr10 awarded Nice Question