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Results tagged with riemannian-geometry
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user 131004
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.
6
votes
1
answer
292
views
Weitzenböck formula and comparison of norms
Let $M$ be a closed Riemannian manifold with a spin$^\mathbb{C}$ bundle $S$. Now for a spin connection $A,$ and a spinor $\phi,$ it can be shown that $C\lvert\nabla_A\phi\rvert^2\geq \lvert D_A\phi\rv …
- 954
3
votes
0
answers
170
views
Bound of the spinor element in Seiberg-Witten equation for a Kähler surface
Let's say we want to solve a perturbed version of SW equations on a closed Kähler manifold $(X,\omega):$
\begin{align*}
&D_A\phi=0\\
&F_A+it\omega=q(\phi)=\phi\otimes\phi^*-\frac{|\phi|^2}{2}\text{Id} …
- 954
1
vote
0
answers
107
views
A Kazhdan-Warner type problem
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 …
- 954
2
votes
0
answers
107
views
A specific question in $G_2$ geometry
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_{ …
- 954
1
vote
1
answer
88
views
A curve in the bundle of two forms
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 …
- 954
2
votes
0
answers
139
views
Is curvature of the canonical line bundle always $(1,1)?$
Let $(M,g,\omega)$ be a symplectic manifold with $g$ and $\omega$ denoting the Riemannian metric and the symplectic form respectively. If $J$ is a compatible almost-complex structure, then is the indu …
- 954
2
votes
0
answers
28
views
Trace-free Hermitian endomorphisms in dimension $7$
Let $M$ be a spin-manifold of dimension $7$. Let $S\rightarrow M$ be a spin-bundle on $M$. Then Clifford multiplication ($c$) gives us the following isomorphism:
\begin{align*}
c:i\Lambda^2\oplus\Lamb …
- 954
3
votes
1
answer
418
views
Pull back of Spin$^{\mathbb{C}}$ bundle
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 …
- 954
5
votes
1
answer
345
views
Understanding the quadratic part in Seiberg Witten equation
Lets take a closed four manifold $M:=\Sigma_1\times \Sigma_2,$ where $\Sigma_i$s are compact Riemann surfaces. Now if $V$ and $W$ are Spin$^\mathbb{C}$ bundles on $\Sigma_1$ and $\Sigma_2$ respectivel …
- 954
5
votes
0
answers
172
views
Curvature of the line bundle $\mathcal{O}(2)$ on the twistor space
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 …
- 954
2
votes
1
answer
192
views
Decomposition of forms on a Spin$(7)$ manifold
Let's take a $G_2$ manifold $(M,\Phi)$, then we get a Spin$(7)$ manifold by taking $(M\times\mathbb{R},\Psi:=\Phi\wedge dt+*_M\Phi),$ where $t$ is the coordinate in the $\mathbb{R}$-direction. $\Phi\i …
- 954
1
vote
0
answers
71
views
Spin(7)-instanton
Let $M$ be a Spin$(7)$-manifold with a spin-bundle $S=S_+\oplus S_-$. There's an obvious connection on $S$ which comes from lifting the Levi-Civita connection. And it induces a connection on the deter …
- 954
3
votes
0
answers
279
views
A question in $\operatorname{Spin}(7)$ geometry
$\DeclareMathOperator\Spin{Spin}$I am looking for a proof of a fact (I think it's true intuitively due to representation theory) in $\Spin(7)$ geometry. Let's take a closed $\Spin(7)$-manifold $(M^8,g …
- 954
6
votes
1
answer
193
views
Reference for $\epsilon$-regularity
I am looking for a reference for the following $\epsilon$-regularity statement: let
$(M,g)$ be a Riemannian manifold of dimension $n$,
$\Delta=dd^*+d^*d$,
$B_r$ denotes a ball of radius $r$ around a …
- 954
4
votes
0
answers
168
views
A question in spin geometry in dimension 8
$\DeclareMathOperator\trace{trace}\DeclareMathOperator\End{End}\DeclareMathOperator\Trace{Trace}$This is to understand a very specific isomorphism in dimension $8$. In dimension $4$ for a spin$^c$ man …
- 954