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Results tagged with riemannian-geometry
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user 47391
Riemannian Geometry is a subfield of Differential Geometry, which specifically studies "Riemannian Manifolds", manifolds with "Riemannian Metrics", which means that they are equipped with continuous inner products.
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(First) Bianchi Identities
I'm looking at a paper of J. A. Thorpe ("Some Remarks on the Gauss-Bonnet Integral"). In the paper he defines "higher-order" notions of curvature. One thinks of the usual curvature tensor as $R\in \La …
- 6,337
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k-th Pontryagin class of $\Lambda^{2k}_{\pm}$ on an oriented $4k$-manifold
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 …
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The Yamabe problem and $\phi^4$ scalar field theory?
The other day I happened to be browsing this page on wikipedia: https://en.wikipedia.org/wiki/Mass_gap
In the middle of the page is the equation $$\square\phi+\lambda\phi^3=0$$ where $\square$ is the …
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1
answer
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"Elliptic" proof that Compact Ricci Solitons are Gradient Ricci Solitons
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 …
- 6,871
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answer
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What does positivity of the first Pontryagin number of a vector bundle tell us?
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 …
- 11.9k
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answer
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A Certain First-Order Differential Equation for a Closed 2-Form
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 …
- 108k
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2
answers
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Converse to Lichnerowicz Vanishing Theorem?
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 …
- 19.4k
6
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2
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Index of Modified Dirac Operator
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 …
- 17.5k
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answer
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Parallel Transport on Hypersurface Spinor Bundle
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 …
- 787
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Parallel Transport on Hypersurface Spinor Bundle
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 …
- 787
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Hamilton's Proof of the Tensor Maximum Principle
My questions come from Richard Hamilton's Three-Manifolds with Positive Ricci Curvature paper. I'm trying to work through parts of the paper so I can better understand the Ricci Flow for my research. …
- 424