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Results tagged with riemannian-geometry
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user 280895
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.
10
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2
answers
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Does the Hodge star operator determine the metric?
Let $M$ be a closed oriented smooth $4$-manifold with two metrics $g$ and $\tilde g$. Consider the Hodge star operators on $2$-forms
$$
\star:\Omega^2(M) \to \Omega^2(M) \quad \text{and} \quad \tilde …
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7
votes
1
answer
202
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Existence of the tubular neighborhood of uniform size
Let $(M^n,g)$ be a compact Riemannian manifold with boundary $\partial M=N$. Suppose $|Rm_g| \le C_1$ on $M$ and the second fundamental form of $N$ is bounded by $C_2$. Moreover, there exists a consta …
- 891
7
votes
1
answer
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Kähler metric with two compatible complex structures
Let $(M^4,g)$ be a complete $4$-dimensional Riemannian manifold such that two almost complex structures $I$ and $J$ are compatible with $g$ and $\nabla_g I=\nabla_g J=0$.
Can we prove that $(M,g)$ is …
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5
votes
1
answer
359
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Lower bound on the first eigenvalue of the Lichnerowicz Laplacian on positive Einstein manif...
Suppose $(M^n,g)$ is an $n$-dimensional Einstein manifold with $Ric=(n-1)g$. Let $\lambda$ be the minimal eigenvalue of the Lichnerowicz Laplacian $\Delta_L$ defined on all transverse-traceless symmet …
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4
votes
1
answer
301
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First eigenvalue of the Laplacian on the traceless-transverse 2-forms
Let $(S^3/\Gamma, g)$ be a spherical space form with constant sectional curvature $1$, where $\Gamma$ is a finite subgroup of $SO(4)$ acting freely on $S^3$.
Consider the first nonzero eigenvalue equa …
- 891
4
votes
1
answer
129
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Rigidity of the compact irreducible symmetric space
Let $(M^n,g)$ be an irreducible symmetric space of compact type. In particular, $(M^n,g)$ is an Einstein manifold with a positive Einstein constant.
Is there any classification for $(M^n,g)$ if $(M^n, …
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3
votes
0
answers
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Ricci deformation of hyperkahler ALE orbifold
Let $(X^2,g)$ be a hyperkahler ALE orbifold surface. Consider its Ricci deformation equation:
$$
\Delta h+2Rm(h)=0
$$
for $\text{div}_g h=\text{Tr}_gh=0$ and $h=O(r^{-\epsilon})$ as $r \to +\infty$. M …
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1
vote
0
answers
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Perturbation of the self-dual harmonic $2$-forms
Let $M$ be a closed oriented smooth $4$-manifold with $m=b^+_2(M)>0$. We set $\mathcal R$ to be the space of smooth metrics on $M$ and consider the following map $f:\mathcal R \to Gr(m,n)$ defined for …
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