# 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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### Principal eigenvalue of non self-adjoint elliptic operators on closed manifolds

Consider the elliptic operator $Lu = - \Delta u + \langle \nabla u , X \rangle + c \, u $ acting on functions on a closed Riemannian manifold $M$. Here $\Delta$ denotes the Laplace-Beltrami operator, $...

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### The convergence of volume form

Let {$g_i$} and $g$ are smooth Riemannian metrics on a closed smooth manifold $M$. Assume $g_i$ converges to $g$ in the $C^ \alpha$-topology, where $\alpha$ may be 0,1, $\infty$ or others, then ...

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### conformal changes to Lorentzian curvature

Let $(M,g)$ be a Lorentzian manifold and let $R$ be the curvature tensor. We say $R\leq 0$ if
$$ g(R(X,Y)Y,X) \leq 0\quad \forall \, X,Y \in TM.$$
My question is whether given a Lorentzian manifold $...

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### minimizing weighted length of closed curves

Let $\mathcal{A}$ be the family of closed smooth curves in the right half of the complex plane $\mathbb{C}$ such that any curve in the family must enlose the point $z=1$ and tangent to the $y$-axis at ...

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### Is the intersection of two distinct sufficiently small metric spheres always empty, a point or a metric sphere of lower dimension?

Let $(X,d)$ be an $n$-dimensional $(n< \infty)$ complete geodesic metric space, where any two points in $X$ are joined by a unique shortest geodesic. Let $S$ be a sufficiently small metric $(n-1)$-...

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### history of geometric mechanics

I was thinking about the foundations of geometric mechanics and its precursors.
I wondered who was the first to realized the equivalence between Riemannian geometry and Lagrangian mechanics. In ...

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### Is the 2-dimensional Gauss-Bonnet theorem applicable in higher dimensions? [closed]

This is a cross-post of this MSE post that users commented that it is appropriate for MO.
I want to know
Question: Is the 2-dimensional Gauss-Bonnet theorem applicable (any topological ...

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### Curvature of nonsymmetric metric tensors?

Consider a smooth manifold $M$ of arbitrary dimension. We have notions of psuedo-Riemannian or Riemannian metrics on a manifold, and they differ in the slightest way of being positive-definite or not. ...

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### Levi-Civita connection from idempotents

Let $(M,g)$ be a closed Riemannian manifold. Let $V$ be a smooth complex vector bundle over $M$. We can write $V$ as the range of an idempotent $E$ in a matrix algebra $M_n(C^\infty(M))$ acting on a ...

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### Lorentzian manifolds of negative spacelike sectional curvature

Suppose $(M,g)$ is a simply connected Lorentzian manifold of signature $(-,+,\ldots,+)$ and such that the sectional curvature of any space-like surface is non-positive. Is it true that there are no ...

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### Reference on area-constrained harmonic maps?

I would appreciate a reference that discusses (at least some of) what is known regarding the existence of constrained harmonic maps from one space into another. More specifically, I am interested in ...

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### Distance function and Hessian in Lorentzian geometries with positive curvature

Suppose $(M,g)$ is a Lorentzian manifold with signature $(-,+,\ldots,+)$ and a positive curvature. Let $p \in M$. Let $U$ be a sufficiently small neighborhood of $p$ in the exterior of the double null ...

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### Quotient distance on $\mathbb{C}P^n$ is equivalent to distance induced by the Fubini-Study metric

Let $n$ be an even, positive integer. Then, is the metric (in the metric-space sense) on $\mathbb{C}P^n$ induced by the Fubini-Study (Riemannian) metric equivalent to the quotient (pseudo?)-metric
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### Length and curvature for closed curves in negatively curved spaces

In the Euclidean plane, for a closed smooth curve of length $\ell$ whose curvature is bounded above by $\epsilon$ we have the inequality
$$ \ell \ge 2\pi \epsilon^{-1} $$
which follows from the fact ...

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### Closed-form expression for Riemannian exponential maps on symmetric spaces

Besides the Poincaré model for the hyperbolic disc, $S^n$, and Euclidean space, what are known instances of a symmetric space $M$ (finite-dimensional) for which the exponential map is known in closed-...

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### Electromagnetic energy in Lovelock gravities

To fix ideas, let us recall that General Relativity describes gravitational phenomena on a 4-dimensional pseudo-Riemannian manifold $(X,g_{ab})$ with field equations that relate the energy-momentum ...

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### Commutation relations between covariant and Lie derivatives

I am currently working on extrinsic riemannian geometry and I am looking for a sort of commutation relation between the covariant and Lie derivatives.
To be more precise : considering an hypersurface ...

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### Totally geodesic submanifolds of SO(3) [closed]

Consider the special orthogonal group $SO(3)$ with its bi-invariant metric (or equivalently, with the metric induced by its standard embedding to the space of $3\times 3$ real matrices).
Obviously, $...

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### Initial value problems on manifolds around submanifolds (reference)

I am looking for a reference on initial value problems formulated on smooth manifolds with initial conditions on submanifolds. More precisely, let $X$ be a smooth manifold and $Y\subset X$ a embedded ...

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### A metric naturally arise from the Euclidean symplectic structure?

For $n>1$ let $\omega=\sum_{i=1}^n dx_i\wedge dy_i$ be the standard symplectic structure on $\mathbb{R}^{2n}=\mathbb{R}^n \times \mathbb{R}^n$.
We define the following distribution $D$ on $\mathbb{...

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### Vector field with constant divergence around embedded submanifold

Let $M$ be a smooth $n$-dimensional manifold and $N\subset M$ be a closed embedded submanifold of codimension at least $2$. Furthermore, let $\mu$ be a volume form on $M$.
Question: Does there ...

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### Is this $1$-form harmonic?

Let $(M^3,g)$ be a compact, connected and oriented Riemannian $3$-manifold with boundary. For a harmonic map $u : M \to \mathbb{S}^1$ satisfying Neumann condition along $\partial M$, let $h = u^*(d \...

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### Continuity/Lipschitz regularity of exponential map from $C_c$ to $\operatorname{Diff}_c$?

For finite-dimensional Lie algebras, see this for a nice example, the exponential map is smooth and in particular, it is locally-Lipschitz onto its image. However, things are different when moving to ...

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### Convergence of free boundary minimal surfaces

I suspect the following statement is true:
Let $(M^3,g)$ be a compact and orientable Riemannian $3$-manifold with nonempty boundary. Let $\{\Sigma_n\}_{n \geq 1}$ be a sequence of compact and ...

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### Surfaces with non-constant negative curvature

Are there any nice models of surfaces with non-constant negative curvature, analogous to the Poincare disk for constant negative curvature. I have found lots of general results and theory but no nice ...

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### Orthogonal divergenceless symmetric tensors

Suppose $ \nabla_{\mu} S^{\mu\nu}=0 $ on a Riemannian manifold with metric $ g $ where $S^{\mu\nu} $ is a symmetric tensor. Given another symmetric tensor constructed from unit vectors satisfying $ ...

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### Reference for “well-known” fact about Laplace-Beltrami on compact manifold?

Every resource I've looked through has always brushed aside the fact that $-\Delta_g$ on a compact manifold without boundary has a discrete spectrum tending to infinity: $0 = \lambda_0 < \lambda_1 \...

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### Is there a diffeomorphism of the disk with constant sum of singular values?

This question is a relaxed version of this question.
Let $D \subseteq \mathbb{R}^2$ be the closed unit disk, and let $c \ge 2$.
Does there exist a diffeomorphism $f:D \to D$ with constant sum of ...

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### Metrics on torus without closed contractible geodesics

It is easy to see that any closed geodesic on a flat 2-torus is noncontractible.
Further the same holds true for a torus of revolution.
Indeed either a closed geodesic is a meridian and therefore ...

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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 ...