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 ...
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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 ...
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$C^2$-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_{...
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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 $...
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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 ...
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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 ...
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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 ...
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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 ...
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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 ...
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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 ...
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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 ...
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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 ...
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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 $q$ ...
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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, ...
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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 ...
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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), ...
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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-...
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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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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 ...
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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 ...
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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 ...
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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#...
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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 $B(p,1)$ ...
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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 ...
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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 ...
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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 ...
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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 $\...
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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 ...
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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 ...
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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 ...
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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 ...
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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 \...
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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 ...
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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 $N$ ...
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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/...
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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 $\theta \in \Lambda^1(TM)$.
$$
d\alpha+\theta \wedge \alpha=0
$$
$$...
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Riemannian manifolds which admit a smooth free $\mathbb{Z}/3\mathbb{Z}$ action but do not admit an equilateral triangle action
A free action of $\mathbb{Z}/3\mathbb{Z}$ on a Riemannian manifold $(M, g)$ is called an equilateral action if for every $x\in M$ all three points of orbit of $x$ have the same distance from each ...
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What does a curve with unbounded acceleration tells about its shape?
Let $(M,g)$ be a compact Riemannian manifold and assume that $c : \mathbb{R} \to M$ is a smooth curve. I am work on a research problem where
$$\left\|\dfrac{\nabla}{dt}c'\right\|\to \infty.$$
What ...
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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 ...
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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 ...
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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 ...
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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 $\tilde{\gamma}$?
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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 ...
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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 $\...
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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 ...
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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 ...
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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 ...
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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\...
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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 $...
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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 ...