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Results tagged with riemannian-geometry
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user 138705
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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Given $Ric_g$ of 3-dim Riemannian manifold, induced metric $\gamma$ and mean curvature $tr_{...
Suppose we have the manifold $\mathbb{R}^3$ equipped with a Riemanian metric $g$ (not necessarily the Euclidean metric. And the induced metric on the $B_1$ (the ball with radius $1$) is $\gamma$.
Su …
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Finding an asymptotically flat manifold with ${\rm Ric}_{r\phi} = \frac{\sin\theta}{r^2}$
Let $(r,\theta,\phi)$ be the spherical coordinates on $\mathbb{R}^3$ where $\theta \in (0,\pi)$ and $\phi\in (0,2\pi)$.
Does there exist an asymptotically flat metric $g$ on $\mathbb{R}^3\setminus B_1 …
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Are metrics of the form $dr^2+ \Omega^2 r^2 g_\text{round}$ asymptotically flat?
Let $M = [1,\infty)\times S^2$. Let $\Omega$ be any smooth positive function on $S^2$.
Is the metric $dr^2+ \Omega^2 r^2 g_\text{round}$ asymptotically flat (where $g_\text{round}$ is the round metric …
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answer
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Finding vector fields on $S^2$ with equal divergence
Let $\mathfrak{X}_{CK}^{\perp}$ be the space of vector fields on $S^2$ that are $L^2$-orthogonal to conformal Killing vector fields. Let $\mathfrak{X}_{CK}$ be the 6-dimensional space of conformal Kil …
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$C^0$ estimate for solutions of elliptic PDE with Neumann BC
Suppose for simplicity that $M$ is a compact n-dimensional submanifold of $\mathbb{R}^n$ with boundary.
Extend $f$ to all $\mathbb{R}^n$ so that it's $0$ outside $M$ and define the function $u_1(x) := …
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Conformal laplacian on asymptotically flat manifolds with boundary
Let $g$ be an asymptotically flat metric on $M = \mathbb{R}^3 \setminus B_1$ where $B_1$ is the unit ball.
Suppose $X$ is a smooth vector field on $M$ that is decaying exponentially and satisfies
$$\D …
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answers
120
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Given the Ricci decays rapidly to 0 at infinity, is the metric asymptotically flat?
Consider the manifold $M=\mathbb{R}^3 \setminus B$ where B is the ball with radius 1. Let $f \in C^{ \infty}(M) $ satisfying:
$$f = \frac{C(\theta, \phi)}{r} + O( r^{-2}) $$
Where $(r,\theta,\phi)$ …
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votes
1
answer
527
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Conformal Killing fields satisfy a third order PDE
Let $(M,g)$ be a $n$-dimensional Riemannian manifold with smooth metric $g$.
Proposition 3.2 in the paper "The Boost Problem in General Relativity" by O'Murchadha and Chistodoulou claims that any conf …
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answers
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Growth/Decay of conformal Killing fields in cone metrics
Let $\gamma$ be a smooth metric on $S^2$ of positive curvature. Consider the metric
$$g= dr^2 + r^2 \gamma$$
on $[1,\infty) \times S^2$.
Does there exist a nontrivial conformal Killing field vanishing …
- 969
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140
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For a 1-parameter family of metrics, how do we compute the derivative of the intrinsic geome...
Consider a family of metrics and functions $(g(t), u(t))$ on $M:= \mathbb{R}^3 \setminus B_1$ satisfying
$$ g(0) = g_0, \quad g'(0) = \tilde g, \quad u(0) = u_0, \quad u'(0) = \tilde u$$
where $g_0$, …
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Finding $\Omega$ such that the 1-form $\Omega^2 \omega$ is $L^2$ orthogonal to conformal kil...
Consider the space $\mathcal{A}$ of functions $\Omega$ such that $\Omega^2 \gamma_0$ is isometric to the round sphere, where $\gamma_0$ is the round sphere. (so $\Omega^2 \gamma_0$ is of constant curv …
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$\Delta_g f = 0$ on the Riemannian Manifold $(\mathbb{R}^3 \setminus B , g)$ with conditions...
Consider the manifold $\mathbb{R}^3 \setminus B$ where $B$ is the ball with radius 1 with riemannian metric $g$ (not necessarily the euclidean metric).
I am looking for solutions to $\Delta_g f = 0$ …
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Divergence of conformal Killing vector fields on $S^2$ and the spherical harmonics
Can anyone think of a conformal Killing vector field $W$ on $S^2$ with the round metric that is not Killing such that its divergence is $L^2$-orthogonal to the spherical harmonics with $\ell = 1$?
One …
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2
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1
answer
223
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The differentiability of the distance function on asymptotically flat manifolds
Let $M = \mathbb{R}^3 \setminus \overline{B_1}$ where $\overline{B_1}$ is the closed unit ball.
Let $g$ be an asymptotically flat metric of the form $g_{ij} = \delta_{ij}+h_{ij}$ in standard coordinat …
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3
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0
answers
156
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Conformal Killing vector fields on manifolds that are not asymptotically flat
Let $M = [1,\infty) \times S^2$.
Equip $M$ with the metric $g = dr^2 + r^2 (\gamma + h)$ where $\gamma$ is a metric on $S^2$ and $h$ is a $(0,2)$ tensor on $M$ that satisfies
$$h = O(1/r),\quad \part …
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