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This is a bit of a vague question, sorry for that. I am wondering if there's any detection of Hodge decomposition in terms of Clifford multiplication. For example if $\phi$ is a spinor and $\theta,\al …

Let $M$ be a closed manifold of dimension $6$, and we look at the collection of smooth functions on $M$ which satisfy: $$ \|f\|_{C^0}\leq C\big(\|f\|_{L^{14}}^2+1\big) $$ for some fixed $C>0$. Can we …

It is known that $S^6$ has (real) killing spinor. I am looking for an explicit description of such a spinor. More generally, we can choose an almost complex structure say, $J$ on $S^6$. Now, we can de …

Let $M$ be an oriented closed $6$-manifold. $V$ be an hermitian complex vector bundle of dimension $4$ on $M.$ Hence $c^3(V)\in H^6(M,\mathbb{Z})\cong \mathbb{Z}$ can be thought of as an integer and o …

We know that Lie Groups are parallelizable, I was looking for a version of the converse and came across this: https://books.google.com/books?id=w4bhBwAAQBAJ&pg=PA115 in Introduction to Smooth Manifold …

Let $X$ be a compact Riemannian manifold and I am interested in the following set of equations: \begin{align*} \Delta f+u\cdot e^{f+\lambda}=c\\ \lambda-2f=g \end{align*} where $u,g$ are given real va …

Let $X$ be a closed manifold. $g:X\rightarrow \mathbb{R}$ be a smooth function ,$\alpha$ a section of a line bundle with discrete zeros and $c>0$ a constant, then Kazdan-Warner's work says that the fo …

It's actually quite easy and I completely missed it. If $f$ is the solution of the original equation: \begin{align*} 2\Delta f+\frac{e^g\lvert\alpha\lvert^2}{4}e^{5f}=c \end{align*} and say $f_\lambd …

It's a well known result by Kazdan and Warner that on a closed Riemannian manifold the pde: \begin{align*} \Delta f+ge^f=c \end{align*} has a unique solution for $g\geq 0,$ and $c$ a positive constant …

Let's take a closed $G_2$ manifold $(M,\Phi)$. $\Phi$ denoting the three-form which defines the $G_2$ structure on $M$. Let's take a closed two form $\theta\in\Omega^2(M).$ Is \begin{align} d(\theta_{ …

Let $(M,g)$ be a closed Riemannian manifold. Fix a point $m\in M$ and a $2$-form $\omega$ at $m.$ Take a curve $\gamma$ in $M$ such that $\gamma(0)=m.$ Now we can get a $2$-form along $\gamma$ by taki …

Let $M$ be a closed $4$-d Riemannian manifold and $Z$ be its twistor space of $M$, i.e., the bundle of almost complex structures on $M$. Let $V$ be a Spin$^{\mathbb{C}}$ bundle, $V_+$ denote the posit …

On any compact Kähler manifold $M^n$ one can ask: given a closed $p,q$ form $\alpha$ on $M$ does $\exists\beta\in\Omega^{p-1,q-1}(M;\mathbb{C})$ such that $\alpha=\partial\bar\partial\beta.$ I am inte …

Say $X$ be a $6$-dimensional compact Riemannian manifold which admits a $Spin^{\mathbb{C}}$ structure. Now I want to have a Clifford multiplication formula by $d\beta$ in terms of $\beta$ where $\beta …

Let $M^4$ be a closed Riemannian manifold and $Z:=S\big(\Lambda^2_+(M)\big)$ denote the twistor space of $M,$ i.e., the sphere bundle of the self-dual 2-forms on $M$. Now at a point $(m,J)\in Z$ the t …