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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Heat flow, decay of the Fisher information, and $\lambda$-displacement convexity

In the whole post I will work in the flat torus $\mathbb T^d=\mathbb R^d/\mathbb Z^d$ and $\rho$ will stand for any probability measure $\mathcal P(\mathbb T^d)$. This question is strongly related to ...
leo monsaingeon's user avatar
1 vote
0 answers
466 views

Horizontal lift of fundamental vector field

Suppose $\theta\colon G\times M\to M$ is a transitive smooth left action of a compact Lie group $G$ on a manifold $M$ and $\pi\colon G\to M\cong G/K$ the corresponding smooth submersion for some ...
Sven Pistre's user avatar
1 vote
0 answers
38 views

$C^2$-control using orthonormal frame on a Riemannian manifold

Let $(M,g)$ be a Riemannian manifold. Let $E=(E_1,\dots,E_n)$ be an orthonormal frame for $M$. So for $M$ itself we have a natural $C^k$-norm $\|f\|_{C^k_g(M)}:=\max\limits_{1\le m\le k}\sup\limits_{...
Liding Yao's user avatar
2 votes
2 answers
106 views

Is the point giving the width in strictly convex surface a cut point?

Assume that $\Sigma$ is a stricly convex surface in $\mathbb{E}^3$ homeomorphic to a sphere. Further, assume that $p_0,\ p_1\in \Sigma$ are intersection points with planes $z=0,\ z=1$ and the surface $...
Hee Kwon Lee's user avatar
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2 votes
0 answers
179 views

The Hopf conjecture for products and slight modification

Perhaps this post won't get too much attention, and I apologize if this is deeply charged with a self perspective more than general facts. I would like to know why do people believe that the standard ...
L.F. Cavenaghi's user avatar
7 votes
0 answers
286 views

Examples of non-compact, holomorphically symplectic Kähler manifolds which are not hyperkähler

Let $(M,\omega_{1},I_{1})$ be a non-compact Kähler manifold. If $M$ admits a holomorphic symplectic form $\Omega$, is it possible M not be hyperkähler? Is there any example? (*)Under the assumption ...
Eder Moraes's user avatar
3 votes
0 answers
130 views

Whitney $C^\infty$ topology for Riemannian Metrics

I'm currently reading the paper "Quadrants of Riemannian Metrics" by Fegan and Millman (https://projecteuclid.org/euclid.mmj/1029002001). In the proof of Proposition 5 at the bottom of page 4, they ...
Logan Clark's user avatar
1 vote
1 answer
256 views

Does there exist a real-valued function on the hyperbolic plane which has bounded hessian norm and unbounded gradient norm?

Does there exist a real-valued function on the hyperbolic plane which has bounded hessian norm and unbounded gradient norm? Specifically, consider the poincare half-plane model of the 2d hyperbolic ...
ccriscitiello's user avatar
4 votes
0 answers
381 views

Intrinsic numerical methods on Riemannian manifolds

I am interested in numerical methods for ordinary differential equations on a Riemannian manifold $M$. The general form of such an equation is $\dot x(t)=V(x(t)), x(0)=x_0 \in M$, where $V$ is a ...
Foivos's user avatar
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-1 votes
1 answer
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A manifold or Riemannian structure on the space of all conjugacy classes of a compact Lie group [closed]

Let $G$ be a compact Lie group. Is each conjugacy class a closed subset of $G$? Define the conjugacy equivalent relation $g\sim h$ if $g$ is conjugate to $h$.Is $G/\sim$ a Haussdoef space with ...
Ali Taghavi's user avatar
15 votes
2 answers
2k views

Riemannian manifold as a metric space

I am looking for a reference to the following simple statement; it must be classical. (It is easy to proof, but I want to have a reference.) A metric space $X$ that corresponds to a Riemannian ...
Anton Petrunin's user avatar
5 votes
1 answer
371 views

Every homotopy class contains at least a harmonic representative

Let $(M^3,g)$ be a closed, connected and oriented Riemannian $3$-manifold. A circle-valued map $v : M \to S^1$ is harmonic iff the gradient $1$-form $\omega_v = v^* d\theta \in \Omega_1(M)$ is ...
Eduardo Longa's user avatar
4 votes
2 answers
217 views

How close are the exponential maps on $\mathbb{S}^2$ at two nearby points?

Consider the two dimensional sphere $\mathbb{S}^2$ and let $p, q \in \mathbb{S}^2$. Let $\text{exp}_{p}$ and $\text{exp}_{q}$ be the exponential maps on $\mathbb{S}^2$ at points $p$ and $q$ ...
April's user avatar
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2 votes
2 answers
210 views

Lower bound for domain of exponential map on Lorentzian manifolds

Let $M$ denote a manifold admitting a Lorentzian metric $g_{ab}$. Essentially, I would like to know the "minimum domain" on which the exponential map is defined at $p\in M$. To make this concrete, ...
user143410's user avatar
12 votes
0 answers
247 views

Jacobi fields on non-geodesic curves

The point of Jacobi fields is to study variations of geodesics through geodesics, but the Jacobi equation $D_t^2 J + R(J,\dot\gamma)\dot\gamma=0$ makes sense for any curve $\gamma$, not just for ...
Ethan Dlugie's user avatar
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6 votes
2 answers
308 views

Quasi-isometric embedding of graphs in non-compact riemannian surfaces

Given a complete riemannian surface $(S,m)$, where $S$ is homeomorphic to $\mathbb{R}^2$, I would like to find a weighted graph $G$ (which means a graph with real non-negative weights on the edges), ...
Louis Esperet's user avatar
4 votes
0 answers
106 views

Stability of bubbles under the heat flow

Let $\Phi : S \times [0,\infty) \to M$ be Struwe's weak global solution to the heat equation with smooth initial data $\phi : S \to M$, where $S$ is a compact surface and $M$ is a compact three-...
James Dibble's user avatar
2 votes
0 answers
427 views

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 $\...
Ali Taghavi's user avatar
1 vote
1 answer
82 views

Assuming the conformal factor is radially decreasing, prove or disprove the uniqueness of geodesic joining origin and points on the boundary of ball

Let $u$ be a radially decreasing function defined on $\mathbb{R}^n$. We consider the metric $g=e^{2u}\delta$ where $\delta$ is the standard Euclidean metric on $\mathbb{R}^n$. Let $B_r$ be the ball ...
student's user avatar
  • 1,320
14 votes
1 answer
645 views

Eigenfunctions of the laplacian on $\mathbb{CP}^n$

I want to find explicit formulas for the eigenfunctions of the Laplacian on $\mathbb{CP}^n$ endowed with the Fubini Study metric. For the first eigenvalue $\lambda_1 = 4(n+1)$, the eigenfunctions ...
freidtchy's user avatar
  • 320
3 votes
0 answers
171 views

References and results for the eigenvalues of Ricci tensor

I am looking for references or results that gives estimates for every eigenvalue of the Ricci tensor. For example, the least eigenvalue is related to the minimum of the Ricci curvature, what can we ...
L.F. Cavenaghi's user avatar
8 votes
1 answer
220 views

Question on Nash's paper on $C^1$ isometric immersions: Why approximating the error tensor $\delta$?

I am trying to go through the classical paper by Nash on the existance of $C^1$ isometric immersion of a Riemannian manifold $(M,g)$ (here is the Jstor link: https://www.jstor.org/stable/1969840?seq=1#...
Nick A.'s user avatar
  • 203
2 votes
0 answers
121 views

Find a manifold with boundary of a geodesic ball being a torus [closed]

I would like to find the answers to the following questions: a. Find a complete $3$-dimensional Riemannian manifold $M$ and a point $p\in M$, such that the boundary of the open geodesic ball $B(p,1)$ ...
Frank Kong's user avatar
0 votes
0 answers
126 views

A Riemannian manifold with a non-degenerate metric and an inner product $u_{\beta} u^{\beta}=1$

The question is: given a Riemannian manifold with a non-degenerate metric g and an inner product $u_{\beta}u^{\beta}=1$, is $\nabla_{\mu} (u_{\alpha}u_{\beta})=0$ without demanding the trivial ...
Kolten's user avatar
  • 9
4 votes
2 answers
449 views

Packing a Riemannian manifold with disjoints balls

Let $M$ be a smooth Riemannian manifold with Riemannian measure $\mu$. I don't suppose that $M$ is complete. Can we find a finite or countable disjoint collection of open (or closed) and relatively ...
Pii_jhi's user avatar
  • 111
2 votes
1 answer
831 views

vector field on the torus [closed]

Is the following statement true? Let $T$ be diffeomorphic to the solid torus. Let $v$ be a vector field such that $v$ and $curl(v)$ are both tangent to $\partial T$ everywhere and $|v|$ is constant ...
J. Doee's user avatar
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1 vote
0 answers
65 views

Why is the first conjugate time continuous on the unit tangent bundle?

Let (M,g) be a complete, connected Riemmanian manifold and SM the unit tangent bundle. Define the map $con:SM\to (0,\infty]$ such that for $v\in SM, con(v)$ is the first positive time such that $\...
Yoda97's user avatar
  • 11
2 votes
1 answer
206 views

When is this differential form harmonic?

Let $(M^3, g)$ be a (closed) Riemannian manifold and let $u: M \to S$ be a harmonic function, where $S$ is a closed orientable surface. If $\omega$ is a $2$-form on $S$, what are sufficient conditions ...
Eduardo Longa's user avatar
4 votes
1 answer
214 views

When is the cut-locus normal coordinate collared

Let $(M,g)$ be a complete $d$-dimensional Riemannian manifold, $p \in M$ be fixed and let $C_p$ be the cut-locus of $p$. Other than when $M$ is non-positively curved (in which $C_p= \emptyset$ by ...
ABIM's user avatar
  • 5,019
2 votes
1 answer
96 views

Characterization of extrinsic distance prevserving embedding (see the definition given!) from low dimensional Euclidean spaces to high dimensions

P.S. I asked the question on MSE more than a week ago, but didn't get any desired answer, so asking here. Let $m < n \in \mathbb{N}$. Let us equip $\mathbb{R}^m, \mathbb{R}^n $ with their ...
Learning math's user avatar
0 votes
0 answers
184 views

Can the Lie derivative of a Riemannian metric be expressed in terms of the Lie derivative of a Lorentzian metric?

On a Lorentzian manifold with metric (M,g) with a vanishing Euler-Poincare characteristic, there exists a line element vector X which has a collinear vector u (Manifold Theory: An introduction for ...
Kolten's user avatar
  • 9
3 votes
1 answer
951 views

Geodesic convexity and the Geometric Hessian

This is an elementary question in differential geometry. We know that for a smooth real-valued function $f$ defined on an open geodesically convex set of a Riemannian manifold $ \mathcal{X} \subset \...
user1952770's user avatar
4 votes
1 answer
260 views

Can every diffeomorphism be rescaled into a volume preserving one?

This is a cross-post. Let $D \subseteq \mathbb{R}^2$ be the closed unit disk, and let $f:D \to D$ be a diffeomorphism. Does there exist a smooth $h \in C^{\infty}(D)$ such that $h\cdot f$ is an ...
Asaf Shachar's user avatar
  • 6,611
4 votes
1 answer
273 views

Injectivity radius of parallel hypersurfaces

Let $(M,g)$ be a Riemannian manifold and let $N$ be a compact hypersurface isometrically embedded into $M$ and let $\eta$ denote a choice of unit normal vector field on $N$. It is then true that $N$ ...
Ryan Vaughn's user avatar
3 votes
1 answer
230 views

Finite-dimensional argument for Morse-Smale pairs?

Using the Sard-Smale theorem, it is relatively easy to show that Morse-Smale pairs on a manifold $M$ (i.e. pairs $(f,g)$ where $g$ is a metric on $M$, $f$ is a Morse function on $M$, and the stable/...
Nikhil Sahoo's user avatar
  • 1,175
2 votes
0 answers
392 views

The Seiberg-Witten equations for forms

I define equations like the Seiberg-Witten equations for forms of a riemannian four-manifold $(M,g)$. $\alpha \in \Lambda^2 (TM)$ and $\theta \in \Lambda^1(TM)$. $$ d\alpha+\theta \wedge \alpha=0 $$ $$...
Antoine Balan's user avatar
4 votes
2 answers
306 views

Riemannian manifolds which admit a smooth free $\mathbb{Z}/3\mathbb{Z}$ action but do not admit an equilateral triangle action

A free action of $\mathbb{Z}/3\mathbb{Z}$ on a Riemannian manifold $(M, g)$ is called an equilateral action if for every $x\in M$ all three points of orbit of $x$ have the same distance from each ...
Ali Taghavi's user avatar
0 votes
0 answers
36 views

What does a curve with unbounded acceleration tells about its shape?

Let $(M,g)$ be a compact Riemannian manifold and assume that $c : \mathbb{R} \to M$ is a smooth curve. I am work on a research problem where $$\left\|\dfrac{\nabla}{dt}c'\right\|\to \infty.$$ What ...
L.F. Cavenaghi's user avatar
1 vote
1 answer
243 views

Injectivity radius with respect to continuous change of metric

Suppose $M$ is a smooth manifold and for each $t\in [0,1]$ let $g_t$ be a Riemannian metric on $M$ such that $t\mapsto g_t$ is continuous. If $(M,g_0)$ has positive injectivity radius, does that imply ...
user153699's user avatar
23 votes
1 answer
744 views

Is every minimal hypersurface in $S^n$ algebraic?

Let $S^n$ be the round n-sphere. Wu-yi Hsiang asked in his paper “Remarks on closed minimal submanifolds in the standard riemannian m-sphere” (1967) the follow question Is every minimal ...
JSCB's user avatar
  • 1,610
6 votes
1 answer
311 views

Integrating the Riemann curvature tensor over a singular 2-disc

There's a classic characterization of the Riemann curvature tensor. Say, take a Riemann metric on an open subset $U$ of $\mathbb R^n$. Given a point $p \in U$ and two vectors $v,w \in T_p U$ you ...
Ryan Budney's user avatar
  • 42.8k
2 votes
1 answer
125 views

Conjugate points depends on choice of geodesic?

Is there a compact Riemannian manifold $M^n$ and $p, q\in M^n$, such that $q$ is conjugate to $p$ along some geodesic $\gamma$, but $q$ is not conjugate to $p$ along another geodesic $\tilde{\gamma}$?
mmaatthh's user avatar
  • 789
16 votes
1 answer
1k views

Is there an area-preserving diffeomorphism of the disk which is nowhere conformal?

This question is a cross-post. Let $D \subseteq \mathbb{R}^2$ be the closed unit disk. Does there exist a smooth area-preserving diffeomorphism $f:D \to D$ that does not have conformal points? I ...
Asaf Shachar's user avatar
  • 6,611
0 votes
0 answers
71 views

Quasi Riemannian submersion and retraction

Let $M, N$ be Riemannian manifolds. A smooth submersion $f:M \to N$ is called a quasi Riemannian submersion if for every $x\in M$ the restriction of linear map $Df_x$ to orthogonal complement of $\...
Ali Taghavi's user avatar
2 votes
1 answer
287 views

Riemannian submanifolds of Euclidean space admitting Lipschitz extension of Lipschitz functions, and converse statement

Let $M \subset \mathbb{R}^p$ be a Riemannian submanifold. In what follows, when we talk about a Lipschitz function $f$ on $M$, namely $f: M \to \mathbb{R}$, we will assume there is a $L > 0$ so ...
Learning math's user avatar
4 votes
1 answer
1k views

Taylor expansion of determinant of Riemannian metric in normal coordinates up to higher order

Let $(M,g)$ be an $n$-dimensional Riemannian manifold. Let $p\in M$, and let $\{x^i\}_{i=1}^n$ be normal coordinates centered around $p$. Using Jacobi field, one can show that the metric $g$ has the ...
Anonymous amateur's user avatar
2 votes
2 answers
1k views

From Riemannian curvature to Ricci curvature in warped product manifold

Let $M=B \times_f F$ be a warped product of two pseudo-Riemannian manifolds. If $X, Y, Z \in L(B)$ and $U, V, W \in L(F)$, (with $L(B)$ and $ L(F)$ I mean the set of all horizontal and vertical lift ...
MathDG's user avatar
  • 242
7 votes
0 answers
196 views

Understanding the odd-dimensional index

Given a Dirac operator $D$ on a closed odd-dimensional manifold $M$, I've sometimes heard it said that the Fredholm index of $D$ vanishes because it is an ungraded self-adjoint operator, so that $\dim\...
geometricK's user avatar
  • 1,851
2 votes
0 answers
71 views

Embeddedness and homology of a limit of minimal surfaces

Consider the following theorem, proved in this paper: Theorem (Theorem 6.1). Suppose we have a sequence $(\Sigma_j, \partial \Sigma_j) \subset (M, \partial M)$ of immersed free boundary minimal $...
Eduardo Longa's user avatar
2 votes
1 answer
317 views

Foliation of tangent bundle arising from exponential map

We first mention our motivation: For $M=\mathbb{R}$ with usual Riemannian metric the exp map $exp:TM\to M$ is in the form$(x,v)\mapsto x+v$ The level sets of this map define a foliation whose leaves ...
Ali Taghavi's user avatar

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