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Complex, contact, Riemannian, pseudo-Riemannian and Finsler geometry, relativity, gauge theory, global analysis.

1 vote
0 answers
98 views

Is a manifold $R^{+}\times X$ with metric $dt^{2}+t^{2}\rm{g}_{X}$ complete?

The cone $C(X)$ over a complete manifold $X$ is defined as $R^{+}\times X$ admits the metric $dt^{2}+t^{2}g_{X}$. The manifold $C(X)$ is conformal to $Cyl(X):=\{R^{+}\times X, dt^{2}+g_{X})$. Followin …
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1 vote
1 answer
179 views

stable bundle on Calabi-Yau 3-fold

Let $E$ be a $G$-bundle over a compact Calabi-Yau 3-fold, $G$ is semi-simple,compact Lie group. If $E$ is a stable bundle, is the first Chern class of $E$ always zero?
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2 votes
0 answers
141 views

Kähler manifold with a global potential

If $(X^{n},\omega)$ is a complete Kähler manifold with a global potential, i.e. $\omega=i\partial\bar{\partial}f$. There are many articles study the $L^{2}$-cohomology of $X$ under some conditions on …
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2 votes
1 answer
208 views

$L^{2}$ Betti number

Let $\tilde{X}$ be a non-compact oriented, Riemannian manifold adimits a smooth metric $\tilde{g}$ on which a discrete group $\Gamma$ of orientation-preserving isometrics acts freely so that the quoti …
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