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If $M^{4k}$ is an oriented Riemannian $4k$-manifold, then the star-operator splits the bundle $\Lambda^{2k}$ into $\pm 1$-eigenspace bundles denoted $\Lambda^{2k}_{\pm}$. I'm curious if anyone has e …

Some context: In the theory of compact, oriented Riemannian Einstein 4-manifolds, there is a a fundamental topological constraint that is implied by the Einstein equations. To wit, if $\chi$ and $\t …

I'm concerned with the following Proposition: If a compact manifold $M$ satisfies $$Rc + \textstyle\frac{1}{2}\mathcal{L}_Xg = \lambda g $$ where $\lambda$ is a constant (i.e. $M$ is a compact Ricci …

Suppose one has an $S^1$ principal bundle $p: P\rightarrow M$, and a closed 2-form $F$ on $M$. Then the pullback form $p^*F$ is closed, vanishes on vertical vectors, and is invariant under the action …

Suppose we have a Riemannian manifold $(M,g)$ and a fixed vector field $X$. Consider the following equations for a 2-form $F$: $$dF=0$$ $$(\delta-\iota_X) F=0$$ Here, $\delta$ is the codifferential i …

The Lichnerowicz vanishing theorem says that if on a compact 4-dimensional spin manifold there exists a metric whose scalar curvature $R>0$, then there are no harmonic spinors; $$D\psi=0 \implies \psi …

Let's say we have an oriented compact 4-d Riemannian spin manifold $(M,g)$. Everybody who's anybody has heard about the index of the Dirac operator $D: S^+\rightarrow S-$; it's the $\hat{A}$-genus, wh …

I now believe that the statement in question (from the paper), "... the Riemannian connection $\bar\nabla$ of $M$... is compatible with $<\,,\,>$ but not with $(\,,\,)$" is false. It seems that both p …

So this has been driving me up a wall. I'm trying to digest parts of the Parker & Taubes paper, "On Witten's Proof of the Positive Energy Theorem." Here's a link: https://projecteuclid.org/download/p …