# 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.

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### One-sided version of the curve-shortening flow

The curve-shortening flow is
$$
\frac{\partial C}{\partial t} = \kappa n
$$
where $\kappa$ is the curvature, and $n$ is the unit normal vector. For a smooth Jordan curve $C\subset\mathbb R^2$ (closed ...

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### Convex hulls of compact sets in a 2-manifold

Let $(\mathbb{R}^2,g)$ be a complete Riemannian manifold. Let $K\subset \mathbb{R}^2$ be a compact, connected set, and let $\text{conv}(K)$ be its convex hull, i.e., the intersection of all ...

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### Jacobi equation and conjugate points on solution curves of the Van der Pol vector field

Let $X$ be a geodesible non vanishing vector field on a manifold $M$. Namely there is a Riemannian structure $(M,g)$ such that all integral curves of $M$ are unparametrized geodesics of the ...

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### What integral formula is being used here?

I am trying to read the paper "Simple closed geodesics on convex surfaces" by E.Calabi and J. Cao and a certain passage is unclear for me. Before, let me contextualize and set up some ...

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### Intersections of geodesics in an "almost flat" plane

Let $g$ be a complete metric on $\mathbb{R}^2$, such that:
Outside of a compact connected set $K\subset \mathbb{R}^2$, the curvature of $g$ vanishes.
The integral of the Gaussian curvature in $K$ is ...

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### Intersection of orbits of earthquake flow on Teichmüller space

Let $\Sigma$ be a closed oriented surface of genus $g\geq2$. We consider $\mu$ and $\nu$ two filling measured laminations on $\Sigma$. (We say that $\mu$ and $\nu$ fill $\Sigma$ if $\Sigma\setminus(\...

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### Riemannian submanifolds of $2$-Wasserstein space

In the article "Wasserstein Geometry Of Gaussian Measures" by Asuka Takatsu the author shows how the space of d-dimensional Gaussian probability measures with non-singular covariance ...

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### Vanishing directional derivatives on $S^2$

Let $u$ be a smooth function defined on the unit sphere $S^2$. Does there exist a plane $P$ passing through the origin such that $P\cap S^2$ contains at least three points $x_1,x_2,x_3$ with $\nabla u(...

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### Decomposition of tensor field on hypersurface

Let $(\mathcal{M},g)$ be a Lorentzian manifold, which is globally of the form $\mathcal{M}\cong I\times\Sigma$, where $I\subset\mathbb{R}$ ("time") and $\Sigma$ ("space") is some $...

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### Closed geodesics on bumpy spheres

Main question:
Does every bumpy Riemannian metric on a sphere have at least three short and prime closed geodesics, for some reasonable definition of short?
E.g., a geodesic $\gamma$ could be called ...

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### The minimal surface operator in a Riemannian metric

Let $\Omega \subset \mathbf{R}^n$ be a domain, and let the cylinder $\Omega \times \mathbf{R}$ above it be endowed with a Riemannian metric $g$. (Note this is not assumed invariant in the vertical ...

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### How to prove the convergence of the functions with the norm above the critical point of Kondrakov?

I have a sequence of smooth functions with norm 1 in $L_q$ space. I need to prove this sequence strongly converges to some function. But I lack the compactness theorem since, $q$ is above the critical ...

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### naturality of the Godbillon-Vey class

This is a problem from Lawrence Conlon's differential manifolds a first course. I do not know how to prove in the following problem
If $f: N \rightarrow M$ is transverse to $\mathcal{F}$, prove that
$$...

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### Example of a curvature with no associated metric

Is there a concrete example of a $4$ tensor $R_{ijkl}$ with the same symmetries as the Riemannian curvature tensor, i.e.
\begin{gather*}
R_{ijkl} = - R_{ijlk},\quad R_{ijkl} = R_{jikl},\quad R_{ijkl} =...

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### Does every ‘curvature’ tensor induce a metric? [duplicate]

So we know that given a Riemannian manifold $(M,g)$ we can calculate its Riemannian curvature $R$. This covariant $4$-tensor field then satisfies some important symmetries
\begin{gather*}
R_{ijkl} = - ...

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### Hadamard submanifolds of $k$-fold product of hyperbolic plane

Let $\kappa>0$ and $d,k$ be a positive integers with $k\ge d$. For $k\in \mathbb{Z}^+$ large enough, can one find a geodesically complete and simply connected $d$-dimensional Riemannian ...

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### Warping a Riemannian manifold until it has non-positive curvature

Let $(M,g)$ be a complete $d$-dimensional Riemannian manifold and let $(\mathbb{H}^2,h)$ be the hyperbolic upper-half plane; and suppose that $\pi_n(M)=\{0\}$ for every $n\in \mathbb{Z}^+$. If $(M,g)$...

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### What is the minimum $n$ for which $\Bbb H^3$ can be isometrically embedded in $\Bbb R^n$ as a bounded set?

Consider the hyperbolic $3$-space $\Bbb H^3$ (i.e., the unique, simply-connected, $3$-dimensional complete Riemannian manifold with a constant negative sectional curvature equal to $-1$). The Nash ...

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### Does the Lie bracket of a certain pair of vector fields vanish?

I'm trying to read section 3 in
J. Jost and Y.L. Xin [JX].
This section has to do with the geometry of Grassmannians. I have a question that comes out of the derivation at the top of page 283 in that ...

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### Ricci flow negative curvature

We denote by $\mathbb{H}^n$ the hyperbolic plane of dimension $n$.
I don't know much about the Ricci flow, so my first question is probably naïve : can one define a normalized Ricci flow such that the ...

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### What is an analogous version of the Ornstein–Uhlenbeck process on Riemannian manifolds?

Recall that the Ornstein–Uhlenbeck (OU) process in $\mathbb{R}^d$ is defined by the following SDE,
$$
d Z_t=\frac{-1}{2} Z_t d t+d W_t, \quad t \geqslant 0
$$
where $\left(W_t\right)_{t \geqslant 0}$ ...

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### Upper bound for the first eigenvalue of the Laplacian on surfaces with boundary

Let $\Sigma$ be a compact smooth surface with boundary. Define
$$\Lambda(\Sigma) := \sup \{ \lambda_1(\Sigma,g) \operatorname{Area}(\Sigma,g) : g \text{ is a smooth Riemannian metric on $\Sigma$} \}$$
...

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### Volume of balls in 3-dimensional manifolds with nonpositive Ricci tensor

The volume of a ball in a 3-dimensional Riemannian manifold with nonpositive Ricci tensor is greater or equal to the volume of an Euclidean ball with the same radius?
It should not be true, but I am ...

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### Does this special vector field affect on sectional curvature?

We have a nowhere vanishing vector field $X$ on a Reimannian manifold $(M,g)$, such that $\mathcal{L}_X g=2\alpha X^\flat \otimes X^\flat$ for a constant non zero $\alpha$. So, for every vector fields ...

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### Bisector of two points in a Riemannian manifold has measure $0$

Let $p,q\in M$, $p\neq q$, where $M$ is a Riemannian manifold. We will let the bisector of $p,q$ be $\mathcal{B}(p,q)=\{x\in M;d(p,x)=d(q,x)\}$. Does $\mathcal{B}(p,q)$ have measure $0$?
I was ...

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### Eigenvalues of the Laplacian on surfaces with boundary

Let $\Sigma$ be a compact smooth surface with boundary. Is it true that the supremum
$$\sup \{ \lambda_1(\Sigma,g) \operatorname{Area}(\Sigma,g) : g \text{ smooth Riemannian metric on $\Sigma$} \}$$
...

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### "Almost" Riemannian submersions with round slices on $S^2\times S^2$

Not so long ago a co-author and I were dealing with conformal changes of Riemannian metrics with the aim of increasing curvature along geodesics. We then observed the following:
For each $S^2_x=\{x\}\...

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### Are there $n$ points dividing a compact Riemannian manifold into equal regions?

Let $M$ be a compact, connected $m$-dimensional Riemannian manifold, and let $n\in\mathbb{N}$. Can we always find distinct points $p_1,\dotsc,p_n\in M$ such that for $i=1,\dotsc,n$ the regions $A_i=\{...

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### Etymology “Kulkarni–Nomizu product”

$\newcommand\KN{\mathbin{\bigcirc\mspace{-20mu}\wedge\mspace{3mu}}}$In the context of (pseudo)-Riemmian geometry, the Kulkarni–Nomizu product is defined to be an operation $\KN$, which takes two ...

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### Existence of a special vector field on Riemannian manifolds?

In a Riemannian Manifold $(M,g)$ a vector field $X$ is said to be Killing vector field if $L_X g$=0 and is said to be conformal if $L_X g= fg$ for some smooth real function $f$ on $M$.
Also, the ...

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### On diffeomorphisms that preserve the metric

Suppose $\Omega\subset \mathbb R^2$ is a bounded domain with smooth boundary and suppose that
$$ F: \Omega \to \Omega,$$
is a diffeomorphism that fixes $\partial \Omega$ (i.e $F|_{\partial \Omega}$ is ...

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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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### Are there Lojasiewicz-Simon estimates with boundary?

Let $M$ be an analytic manifold with boundary $\partial M$, equipped with a Riemannian metric $g$, which is also analytic up to and including the boundary.
Are there Lojasiewicz–Simon estimates ...

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### Variation of the Einstein Hilbert action in a coordinate-free way

I am currently writing an expository paper on gauge theory and gravity and throughout the gauge theory part I have been able to mostly stay away from coordinates unless I wished to
provide specific ...

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### How to extend this PDE?

Let $(M^n,g)$ and $(N^m,h)$ be Riemann manifolds without boundary of dimension $n$ and $m$ respectively and $u:(M^n,g)\to (N^m,h)$ be a map satisfying the following PDE on $M^n\backslash\Sigma$ ($u$ ...

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### Maximum symmetry metric on Cayley plane $ F_4/{\operatorname{Spin}(9)}$

The maximum symmetry metric on real projective space $ \mathbb{RP}^n $ is the round metric.
The maximum symmetry metric on complex projective space $ \mathbb{CP}^n $ is the Fubini–Study metric; see ...

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### Local Lipschitz constant of exponential map for Hadamard manifolds

Suppose that $(M,X)$ is a simply connected complete Riemannian manifold with pinched sectional curvature between $[a,0]$. Let $r>0$ and fix any point $p\in M$. Is there a bound on the local ...

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### Equivalence between two Sobolev norms on manifolds

On a compact Riemannian manifold $(M,g)$ without boundary, there are two ways to define a Sobolev norm on $M$. Assume that $f\in C^\infty(M)$ in the following.
Use pseudo-differential operators on $M$...

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### Sobolev embedding on sphere

Let $S$ be a two-dimensional sphere, $\Delta$ be the Laplace-Beltrami operator on $S$ and $L^p(S)$, $p\geq 1$, be the usual $L^p$ space of real-valued functions on $S$. We also set $\|f\|_{H^\alpha(S)}...

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### Geometric inequality related with convexity of the boundary

I'm new to Mathoverflow, so hopefully my question is well-posed.
My problem states as follows:
Let $\Omega\subset\mathbb{R}^n$ be a bounded Lipschitz domain with boundary $\partial \Omega$ , $\delta&...

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### If a Dirichlet problem is solved by transforming into ODE (proving its radial symmetry) how can we study it on manifold?

If a Dirichlet problem (elliptic PDE, in $R^{n}$) is solved by transforming into ODE (proving its radial symmetry) how can we study it on manifold?
For example, $B$ is the unit ball in $R^{n}$, the ...

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### About the liouville equation $\Delta u = - \lambda e ^{u}$ on compact manifold with dimension $>2 $

I want to ask about the liouville equation $\Delta u = - \lambda e ^{u}$ on compact manifold with dimension $>2 $.
There are many studies on this equation on Riemannian surface (dimension = 2), for ...

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### Checking that the image of a curve is not contained in a hyperplane

Let $\gamma : [0,1] \to \mathbb R^n$ be a smooth curve, $n \geq 2$. I would like to find an easy to check condition such that the image of $\gamma$ is not contained in an $n-1$ dimensional linear ...

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### Weitzenbock- Anti-selfdual

In "The Theory of Gauge Fields in Four Manifolds", B.Lawson proves the Bochner-Weitzenbock, for an anti-self-dual field $\Psi \in \Omega^2_-(\mathfrak{G}_E)$,where $\mathfrak{G}_E$ is the ...

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### Are geodesics necessarily embedded?

I would like to ask a very basic/naive question. Given a Riemannian or pseudo-Riemannian manidold equipped with the Levi-Civita connection, is it known that all solutions of the geodesics equation are ...

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### The space of Sobolev maps between Riemannian manifolds

Let $\mathcal{M}, \mathcal{N}$ be two Riemannian manifods. Suppose that $\mathcal{N}$ is properly and isometrically embedded in $\mathbb{R}^n$. The space of Sobolev maps between $\mathcal{M}$ and $\...

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### What's the limit of a sequence of harmonic maps between manifolds?

Let ${M}, \, {N}$ be two Riemannian manifolds, and let $u_n: {M} \to {N}$ be a sequence of harmonic maps.
Question. Suppose that $u_n$ converges uniformly to a (necessarily continuous) function $u$. ...

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### Elworthy’s 1982 “Stochastic Differential Equations on Manifolds” - relevant?

In 1982, D. Elworthy published “Stochastic Differential Equations on Manifolds”. Apparently, this was quite a seminal book in the field of stochastic DE’s/processes on manifolds. Is this reference ...

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### Can the exponential map be used to define geodesics (and hence, generalisations of geodesics)?

Let $(M,g)$ be a (connected, paracompact, $C^{\infty}$-smooth) Riemannian manifold with Riemannian metric $g$. The exponential map is defined for each point $p \in M$ to be the map $\exp_p : T_p M \to ...

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### Consequences of integral condition for a vector field

On a compact Riemannian manifold without boundary, if $X$ is a vector field with $\int_M X(fdg-gdf)=0$ for all smooth functions $f,g$, then what can be said of $X$? Taking $f$ constant shows $X$ has ...