Questions tagged [riemannian-geometry]
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.
3,083 questions
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How to calculate the infimum of Yamabe functional on upper hemisphere
We introduce the following functional to study Yamabe problem with boundary.
$$
Q_g(\varphi)=\frac{\int_M\left(|\nabla \varphi|_g^2+\frac{n-2}{4(n-1)} R_g \varphi^2\right) d v+\frac{n-2}{2} \int_{\...
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Request for two unpublished articles of Detlef Gromoll
I am looking for two articles for my research purpose, which are entitled with "Convex riemannian manifolds" and "Convex sets in riemannian manifolds" by Detlef Gromoll.
I would ...
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Taylor series on a Riemannian manifold
I need some help for the following problem.
Let $M$ a riemannian manifold and $f$ a smooth differential function, then consider the following integral $$\int_M \Gamma(x,y)(f(y)-f(x))dV_y$$
where $dV_y$...
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Local geometry of nonumbilic points
I'm reading Escobar's The Yamabe Problem On Manifolds With Boundary.
He says
Let $(y_{1},\cdots,y_{n})$ be normal coordinates around $0\in \partial M$, such that $\eta(0)=-\frac{\partial}{\partial ...
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Symmetric spaces are quandles. Is this important?
For concreteness, let's consider a connected reductive Lie group $G$, and an involution $\theta$ on it. Then the associated symmetric space $X=G/G^\theta$ has the structure of an involutive quandle: ...
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Is the standard $\mathbb R^4$ the only one with positive sectional curvature?
Perelman--Cheeger--Gromoll Soul Theorem states that whenever a complete non-compact Riemannian manifold $(M,g)$ has positive sectional curvature, it should be diffeomorphic to an Euclidean Space.
On ...
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Screened Poisson equation
The screened Poisson equation, i.e.
$$
[\nabla^2−\lambda^2]\phi(r) =
−\psi(r),$$
occurs frequently in physics, including Yukawa theory of
mesons, in electric field screening in plasmas and nonlocal ...
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Pair of laminations that fill on a closed surface
Let $S$ be a hyperbolic surface of genus $g \geq 2$.
A discrete geodesic lamination on $S$ is a set of disjoint, simple, closed geodesics.
Let $L_{1}$ and $L_{2}$ be two discrete geodesic laminations ...
- 28.2k
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Ricci scalar of submanifold of $\mathbf R^n$
Let $M$ be an $n-m$ dimensional sub-manifold of $\mathbf R^n$ defined by the following set of equations:
\begin{equation}
f_1(\vec x)=0, \\
\vdots \\
f_m(\vec x)=0,
\end{equation}
where $\vec x$ are ...
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Volume of submanifold as integral of delta-function
Let $M$ be an $n-m$ dimensional sub-manifold of $\mathbf R^n$ defined by the following set of equations:
\begin{equation}
f_1(\vec x)=0, \\
\vdots \\
f_m(\vec x)=0,
\end{equation}
(where $\vec x$ are ...
- 521
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Complex quadric as a symmetric space
It is known that a smooth complex quadric is a symmetric space. For example, it is
$$\operatorname{Spin}(n+2)/G$$
where $G$ is the maximal parabolic subgroup.
I want a reference for more details and ...
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When are the Schoen-Yau minimal surfaces embedded?
In 1979, Rick Schoen and Shing-Tung Yau published an Annals paper in which they proved the existence of so-called 'incompressible' minimal surfaces in compact Riemannian manifolds.
Question. Under ...
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Volume of sub-manifold of $\mathbf R^n$
Let $M$ be an $n-m$ dimensional sub-manifold of $\mathbf R^n$ defined by the following set of polynomial equations:
\begin{equation}
P_1(\vec x)=0, \\
\vdots \\
P_m(\vec x)=0,
\end{equation}
(where $\...
- 6,367
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Reference for shortest educational path to (Riemannian) hyperbolic plane
I am teaching an undergraduate class for math majors on axiomatic geometry, culminating in the proof that hyperbolic geometry is equiconsistent* with Euclidean geometry. I would like to make an end-of-...
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Showing bound $\|\nabla_t \dot{\tilde{x}}_Y(h, 0)\| \le L \|Y\|_{2, \infty}$ for smooth homotopies of geodesics
This question pertains to Lemma 3.5 of this article. Let $M$ be a smooth Riemannian manifold and $\gamma$ some geodesic with respect to the Levi-Civita connection $\nabla$. For any $C^2$ vector field $...
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A question on convexity and conjugate points
Let $(M,g)$ be a compact smooth simply connected Riemannian manifold with a smooth boundary. Assume also that $(M,g)$ does not have any conjugate pairs of points. Let $\Gamma \subset \partial M$ be a ...
- 5,048
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On properties of Besse spheres
Let $(\mathbb{S}^2,g)$ be a Besse sphere, that is, a Riemannian sphere all of whose geodesics are closed. By a result of Gromoll and Grove, all the geodesics are simple (no self-intersections) and ...
- 5,048
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Does the isometry group determine the Riemannian metric?
Suppose $G \subset \text{Iso}(M)$ is a Lie group acting smoothly on a (pseudo-)Riemannian manifold $(M, g)$. Then $G$ induces a Lie algebra of Killing vectors on $M$. In this paper by Goenner and ...
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Intuition behind the notion of distance between curves
Let $(M,g)$ be a Riemannian manifold and let $p$ and $q$ be two points on it and define $d(p,q)$ as the length of the minimizing geodesic between them. Now given two rectifiable paths $\gamma_1$ and $\...
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Seek "typical examples" for the structure of spaces with two-sided Ricci bounds
By a 1990 paper of Michael Anderson, the following is true:
Theorem. Let the metric space $(X,d,p)$ be a pointed Gromov-Hausdorff limit of a sequence of complete pointed Riemannian manifolds $(M_i,...
- 2,821
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Grassmannian of oriented real $k$-planes
The Grassmann manifold $\widetilde{Gr}(k,\Bbb{R}^n)$ of oriented $k$-planes in $\Bbb{R}^n$ is a double cover of the Grassmann manifold $Gr(k,\Bbb{R}^n)$ of non-oriented $k$-planes. We can give $\...
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John Nash's Mathematical Legacy
It would seem that John Nash and his wife Alicia died tragically in a car accident on May 23, 2015 (reference). My condolences to his family and friends.
Maybe this is an appropriate time to ask a ...
CommunityBot
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$(1+\epsilon)$-bilipschitz parametrization of Lipschitz manifold
Let $\mathscr{H}^m$ be the $m$ dimensional Hausdorff measure in $\mathbb{R}^n$, $m\leq n$. Is it true that for $\mathscr{H}^m$-almost every point $p$ on a Lipschitz manifold $M$ of dimension $m$ ...
- 1,149
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Construct a hypersurface with fixed principal curvatures at a point
I'm reading Eschenburg's paper Local convexity and nonnegative curvature —
Gromov's proof of the sphere theorem recently. And I meet a little question: Given a point $p\in M$, $N\in T_pM$, we want to ...
- 27.5k
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Geodesics and gradient flow
Is there a construction in Riemannian geometry which relates the gradient flow of a function on a manifold with a certain metric with geodesics on another related manifold with its own metric?
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Smoothness of boundary of $r$-neighborhood of convex core
The boundary of an $r$-neighborhood of the convex core of hyperbolic $n$-manifold is smooth, by page 73 of Hyperbolic Manifolds and Kleinian Groups. The authors does not provide a proof for this fact. ...
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Smooth circle actions on Riemannian manifolds and harmonicity of quotient map
Let $(M, g)$ be a compact Riemannian manifold. Suppose $M$ admits a smooth free circle action (Denote the circle group by $G$. The action $G$ on $M$ is not necessarily isometric) and the orbit space $...
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Geometric evolution of convex surfaces to a round sphere
Let $𝑀 = 𝑀^2$ be an embedded convex surface in $\mathbb R^3$ and let $𝑁 ∶ 𝑀 → 𝕊^2$ be the Gauss map for $𝑀.$ Let $𝑉_𝑀$ be the area measure on $𝑀$ and $𝑁_∗𝑉_𝑀$ the corresponding pushforward ...
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English translation of paper: Sur les variétés riemanniennes à groupe d'holonomie G2 ou Spin(7)
I am interested in the history of $G_2$ manifolds and want to read this paper in english:
Sur les variétés riemanniennes à groupe d'holonomie G2 ou Spin(7)
Does anyone know where I can find a ...
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Under what conditions can an orientable Riemannian 3-manifold be defined implicitly?
Under what conditions can an orientable Riemannian 3-manifold $\Sigma$ be defined implicitly?
What I mean by implicitly is that there exists a smooth function $f:\mathbb{R}^n\to \mathbb{R}^m$, such ...
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Curvature explosion and metric landmark stability
$\newcommand{\prin}{\mathrm{prin}}$Context: Let $S$ be the unit sphere in some finite dimensional vector space $V$. Given a connected compact Lie group real representation $\rho:G\rightarrow O(V)$, ...
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Intersection of self-shrinkers
I have a problem regarding a statement in the paper Smooth compactness of self-shrinkers by Colding and Minicozzi.
In the article, they define a surface $\Sigma$ in $\mathbb R^3$ to be a self-shrinker ...
CommunityBot
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A property of almost Riemannian submersions
Definition. A smooth map $f\colon M\to N$ of two smooth Riemannian manifolds is called $\varepsilon$-almost Riemannian submersion (here $0\leq \varepsilon<1/100$) if for any point $x\in M$ and any ...
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Isometry induced on the Grassmannian of planes to the tangent space
Let $(M,g)$ be a Riemannian manifold and assume that $G$ acts isometrically and effectively on $M$. We can then split $TM = \mathcal V\oplus \mathcal H$ globally, though it not does not need to ...
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What is the Freudenthal compactification of a wildly punctured n-sphere?
Let $C$ be a compact and totally-disconnected subspace of the $n$-sphere $\mathbb{S}^n$, where $n\geq 2$.
Question: Must the Freudenthal compactification of $\mathbb{S}^n \setminus C$ be homeomorphic ...
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Computation of scalar curvature from a Riemannian metric
I want to compute the scalar curvature for points on an empirical manifold (sampled data).
I have already an algorithm that learns the Riemannian metric and computes geodesics, so from the metric I ...
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A Fourier elliptic vector field on a Riemannian manifold
Motivation for this question:
Let $X$ be a vector field on a manifold $M$. Obviously the differential operator $D:C^{\infty}(M)\to C^{\infty}(M)$ with $D(f)=X.f$ is not an elliptic opetator when $\...
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Space filling curves
The classic Hahn-Mazurkiewicz theorem has the following consequence: Let $X$ be a compact, connected topological manifold. Then there is a continuous surjective map $f: [0,1] \rightarrow X$.
It is ...
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Understanding the proof of lemma 1.1 from Fisher, Marsden, and Moncrief's paper
The following lemma is from Fisher, Marsden, and Moncrief's paper: the structure of the space of solutions of Einstein's equations:1
1.1. Lemma.
If Ein( $\left.{ }^{(4)} g\right)=0$, and ${ }^{(4)} h$ ...
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Does every smooth manifold of infinite topological type admit a complete Riemannian metric?
To elaborate a bit, I should say that the question of the existence of a complete metric is only of interest in the case of manifolds of infinite topological type; if a manifold is compact, any metric ...
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Do all compact manifolds admit geodesic tiling
Let $M$ be a compact Riemannian manifold. I'll call a set of non-empty subsets $C_1,\dots,C_N$ a geodesic tiling of $M$ if:
Each $C_n$ is closed (geodesically) convex hull of a finite number of $\{...
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Can the hyperbolic plane be immersed in three dimensional Euclidean space, if we are only looking for a weak solution?
Consider the following question:
"Can the hyperbolic plane $(\mathbb{R}^2, g_H)$ be isometrically
immersed in three dimensional Eulidean space$(\mathbb{R}^3, g_{flat})$?"
I believe the answer to ...
- 4,713
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Image of tori in locally symmetric spaces and homology
Suppose we have a reductive group $G$ over $\mathbb{Q}$, a compact subgroup of the adelic points $K_f\subset G(\mathbb{A}_{\mathbb{Q}})$, and the associated locally symmetric space
$$Y_K := G(\mathbb{...
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Information about Milnor conjecture
I'm a student of mathematics and I need know about the status of the Milnor conjecture (if there are partial results or if someone solved that). The statement is:
A complete Riemannian manifold with ...
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Local diagonalisation of a degenerated 2d metric tensor
Consider a smooth 2d-manifold $M$ and let $g$ be a smooth $(0,2)$-tensor satisfying $rk(g)\geq1$ everywhere. Obviously if $rk(g)=2$ at a point $p\in M$ then $g$ is locally diagonalisable (i.e. there ...
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Planar curves in $M^{m}$ vs curves in $M^{2}$
Following Anton Petrunin’s suggestion, I revise the question to make it less vague.
Let $M^{m}$ be an $m$-dimensional Riemannian manifold, and let $\gamma$ be a unit-speed curve $I \to M^{m}$. We say ...
- 108k
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Question about the second order linear elliptic PDE on closed manifold
Recently I see a question
linear second order PDE
in which user Pedro post a reference in Gilbarg's book, which said that the solvability of the linear PDE
$$
\Delta u +B^{i}(x)u_{i}+C(x)u=f
$$
is ...
- 809
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Extrinsically flat submanifolds of a Riemannian manifold
Let $Q$ be a $d$-dimensional Riemannian manifold. A submanifold $M$ of $Q$ is said to be extrinsically flat if $R_{M}(X,Y,Z,W) = R_{Q}(X,Y,Z,W)$ for all $X,Y,Z,W \in \mathfrak{X}(M)$, where $R_{M}$ ...
- 108k
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To what extent is the Nash embedding not unique?
Consider a smooth Nash embedding, $f$, of a Riemannian manifold $Σ$ into Euclidean space $\mathbb R^n$. To what extent is this embedding not unique?
It is clear that the set of all such embeddings ...
- 521
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Naturality of geodesic flow
Let $\texttt{Man}$ be the category of smooth manifolds with local diffeomorphisms as morphisms, and $ \texttt{Bun}$ --- the category of bundles (affine bundles or just fibre bundles, if necessary) and ...
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