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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rigidity of eigenvalues of circular ensemble

Given a circular unitary ensemble, with the following joint density: $p(\theta_1,\ldots, \theta_n) = Z_n \prod_{j < k} |e^{i \theta_j} - e^{i \theta_k}|^2$, is the following statement true? With ...
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Minimal surfaces + Semi-Geodesic Coordinates

Let $(M,g)$ be a three dimensional smooth Riemannian manifold and suppose that $\Gamma$ is an embedded minimal surface in $M$. Define the Fermi or semigeodesic coordinates around this surface through ...
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Isothermal-related functions in higher dimensions

I am interested in getting some geometrical or analytical perspective in studing the following complex pde. I would appreciate any help. Consider $ (M,g)$ to be a 3 dimensional Riemannian manifold ...
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Obstructions for existence of a Riemannian metric such that a given function is harmonic

Let $f:\mathbb{R}^{n}\to \mathbb{R}$ be a smooth function. What type of obstructions exist for existence of a Riemannian metric $g$ on $\mathbb{R}^{n}$ such that $f$ is a harmonic ...
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66 views

Frobenius Condition for a specific first order pde

I would appreciate it if Someone would be kind enough to share some insights about the following question: Suppose $(M,g)$ is a 3 dimensional Riemannian manifold. Consider the following system of ...
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101 views

Handle body of 3-manifold with boundary

We know from Morse theory that smooth manifold(with or without boundary) is a handlebody. However, I found a paper "Three-dimensional manifolds with boundary of nonnegative Ricci curvature" by Ananov, ...
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103 views

Averaging maps of Riemannian manifolds

Let $M$ be a compact Riemannian manifold. We know how to average functions $f\colon M\to {\mathbb R}$; the integral $\frac{\int_M f}{\int_M 1}$ returns a value in ${\mathbb R}$. If intead $f\colon ...
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How large can you draw an island on a map?

A cartographer friend asked me this question: could you classify (shapes of) islands by how much space they occupy on a map (comparatively to how much space is occupied by water) if you draw them as ...
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Flat manifolds and irreducible representations

Let $M$ be a compact Riemannian manifold with vanishing curvature of Levi-Civita connection. Such manifolds were classified by Bieberbach; sometimes they are called Bieberbach manifolds. According to ...
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1answer
85 views

Relationship between Laplacian and Hessian on compact Lie groups

If $f: \mathbb{R}^n \rightarrow \mathbb{R}$ is smooth and compactly supported, one has $$\int |\Delta f(\mathbf{x})|^2\,d\mathbf{x} = \int \| Hf(\mathbf{x}) \|_F^2\,d\mathbf{x}\,,$$ where $\Delta$ ...
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Existence, uniqueness, and smoothness of a solution to a first order PDE on Riemannian M

Let $\mathcal{M}$ be an $n$-dimensional compact Riemannian manifold, and let $\mathcal{A} \subset \mathcal{M}$ be an $n$-dimensional Riemannian submanifold. I wish to determine the local existence, ...
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Bi-Lipschitz classification of germs of conformal metrics at a singularity

First let me introduce some definitions. By a germ of conformal metrics at a singularity, or simply a germ, I mean a conformal Riemannian metric $g$ defined on a punctured neighborhood $U$ of $0$ in ...
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Is displacement controled by stable norm?

Let $T^n$ be the $n$-dimensional torus and $g$ be a Riemannian metric on $T^n$. Let $\tilde g$ be the induced metric on the universal covering; using suitable coordinates, $\tilde g$ is therefore a ...
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197 views

Isometries of hyper-Kähler manifolds

For the purposes of this question, a hyper-Kähler manifold will be a complete connected Riemannian manifold $(\mathcal{M},g)$ whose holonomy representation is isomorphic to the natural representation ...
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1answer
148 views

Divergence invariant lifting of a vector field via a submersion

What is an example of a smooth submersion $P:S^{3}\to S^{2}$ for which the following statment is Not true: For every vector field $X$ on $S^{2}$ there is a non vanishing vector field ...
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Triangle area on surfaces of constant curvature

I am looking for an elementary derivation of the formula for the area of a geodesic triangle lying in a surface of constant curvature $\kappa$, depending on the angles and side length. Of course, the ...
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Synthetic vs. classical differential geometry

To provide context, I'm a differential geometry grad student from a physics background. I know some category theory (at the level of Simmons) and differential and Riemannian geometry (at the level of ...
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1answer
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Gaussian Curvature of Exponentiated 2-Planes

Consider a Riemannian manifold $M$ with sectional curvatures $K\ge 0$ and let $\Pi$ be a 2-plane in the tangent space of $M$ at a point $p$. In a small enough neighborhood $U$ of 0 the exponential map ...
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103 views

Green's function and eigenvalues with multiplicity

Green's function of a differential operator contains a lot of information of that operator. In particular, if we have a differential operator on a compact manifold with discrete spectrum, then Green's ...
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122 views

Why does Hodge decomposition fail in the pseudo-Riemannian case?

Why does Hodge decomposition fail in the pseudo-Riemannian case? Does there exist a special class of pseudo-Riemannian manifolds for which it does not fail, for example Lie groups?
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3answers
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Number of disjoint simple closed geodesics

According to Jairo comment on the first version of this question I revise the question as follows; Let $g$ be a real analytic Riemannan metric on $S^{2}$. Is it true to say that: There are at most a ...
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1answer
486 views

Thurston geometries in dimension 4

In the sense of W. Thurston here, there is 3 geometries in dimension 2 and there is 8 geometries in dimension 3. Question: How many different geometries (in the sense of Thurston) do we have in ...
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2answers
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Can simply or not simply connected maximally symmetric (Semi-)Riemannian manifold be completely classified?

A m-dimensional completed and connected (Semi-)Riemannian manifold which has $m(m+1)/2$ independent global Killing vector fields is called maximally symmetric space. Then what are all possibilities ...
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Is the heat kernel more spread out with a smaller metric?

Suppose M is a smooth manifold, and we have two Riemannian metrics on M, say g and h, with g bigger than h (i.e. for every tangent vector at every point, the norm according to g is bigger than the ...
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Does this squared distance functional have a unique critical point on geodesically convex manifolds?

Let $M$ be a Riemannian manifold with distance function $d$, $C \subset M$ a geodesically convex set, $a=(a_i)_{i=1}^n \in C^n$, $W \in \mathbb{R}_{\geq 0}^{n \times n}$ and $J\colon C^n \rightarrow ...
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Question on a paper of Schoen and Yau

I am trying to understand the paper "Conformally flat manifolds, Kleinian groups and scalar curvature" by Schoen and Yau. In P.56, it says: This implies that $\partial M$ has a zero $q$-capacity, ...
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Is there some Riemannian manifold's version of Whitney theorem?

Given any Riemannian or Semi-Riemannian manifold $(M,g)$, does there exist a Eucildean space $(E,g^\prime)$ of enough high dimension with metric $g^\prime=diag\{-1,-1,...,+1,+1,...\}$ with any n ...
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1answer
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Riemann's formula for the metric in a normal neighborhood

I would love to understand the famous formula $g_{ij}(x) = \delta_{ij} + \frac{1}{3}R_{kijl}x^kx^l +O(||x||^3)$, which is valid in Riemannian normal coordinates and possibly more general situations. ...
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2answers
686 views

When is the Riemannian manifold $\mathbb{R}^n$ complete as a metric space with respect to the Riemannian distance?

Consider the Riemannian manifold $\mathbb{R}^n$ and a smooth Riemannian metric $G:\mathbb{R}^n\rightarrow\mathbb{R}^{n\times{n}}$. What is the minimum assumption on $G$ such that the manifold ...
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442 views

A metric on $S^{2}$ [closed]

Edit:Can this new version of this question be answered with the method of same comments to the previous version? Let $p:S^{3}\to S^{2}$ be the Hopf fibration $p(z,w)= (\parallel ...
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Surface with bounded torsion propery [closed]

Let $S$ be a surface in $\mathbb{R}^{3}$ with the following property: There is a uniform constant $M$ such that for every Frenet curve $\gamma(t)$, contained in $S$, $| \tau(t) | \leq M$, for all ...
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Uniqueness of scalar curvature

I'm reading Gromov's notes http://www.ihes.fr/~gromov/topics/SpacesandQuestions.pdf and at page 7 they say that there is a unique second order differential operator $S$ from the space of Riemannian ...
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1answer
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What is the expression of first eigen function of Laplacian on Hyperbolic plane?

Let $\Delta$ be the Laplacian (a positive operator) on $H^2$ the hyperbolic plane. My question is what is the expression of the eigenfunction $\Delta f= f/4$? (say in the ploar coordiante)
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Is there any progress on Problem 13 (from Schoen and Yau)?

This is closed related to the question asked here. I wonder if there is any progress on Problem 13 from the "Problem Section" in Schoen and Yau, page 281, problem 13, which asks: Let $M_1$ and $M_2$ ...
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1answer
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The complex heat kernel on a Riemann manifold

There is a vast literature available for the heat kernel. Nevertheless, I haven't been able to find almost anything useful about the kernel of the equation $\frac{1}{\mathbb{i}} \frac{\partial ...
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Generalized metric on spacetimes

I read many articles about space-times. Most authors consider these spaces as warped product manifolds $I\times M$ where $I$ is an open connected interval of the real line and $M$ is a Riemannian ...
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1answer
111 views

Has uniform ellipticity implications on the spectrum?

Let $X$ be a complete Riemann surface with a smooth metric, and $L$ a line bundle on it also equipped with a smooth metric; associated to this data there is a Laplace-Beltrami operator $D_L$ acting on ...
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2answers
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Limit cycles as closed geodesics(geodesible flow)

The classical Van der Pol equation is the following vector field on $\mathbb{R}^{2}$: \begin{equation}\cases{\dot{x}=y-(x^{3}-x)\\ \dot{y}=-x}\end{equation} This equation defines a foliation on ...
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3answers
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Manifolds admitting flat connections

For each Riemannian manifold one can construct the Levi-Civita connection. While this connection is unique, we can call a (Riemannian) manifold flat if the Levi-Civita connection is flat. However when ...
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Does local reducibility imply global reducibility of universal covering?

Let $M$ be a locally reducible complete Riemannian manifold, that is, for any $p \in M$, we can find an open set $U$ around $p$ and two Riemannian manifolds $X$ and $Y$ such that $U$ is isometric to ...
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Harmonic maps and centers of mass in Riemannian manifolds

Consider a smooth map $f : M \to N$ between two Riemannian manifolds $(M,g)$ and $(N,h)$. I would like to think of the tension field of $f$ and the harmonicity of $f$ in terms of centers of mass. I ...
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2answers
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Reference for when a metric on a four-manifold is Kahler?

In a paper of Derdzinski (Proposition 5), he proved that if $\delta W^+=0$ and $W^+$ has at most two distinct eigenvalues, then the metric is (locally) conformally Kahler, and if in addition the ...
3
votes
1answer
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Horizontal lift of differential operator

On a Riemannian manifold $M$, there is a canonical horizontal lift $X^{\mathrm{hor}}$ of vector fields $X$ to $TM$, which is characterized by the two properties that $X^{\mathrm{hor}}$ is a ...
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Generalized Hawking Mass

This is a fairly general question. Let $(M^3,g)$ be a Riemannian 3-manifold. Let $\Sigma^2$ be a dimension-2 submanifold of $M$. The Hawking mass of $\Sigma^2$ is defined as $m(\Sigma^2) := ...
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Hilbert's Theorem relevance to positive curvature

In differential geometry, Hilbert's theorem (1901) states that there exists no complete regular surface S of constant negative Gaussian curvature K immersed in $ R^3 $. This theorem answers the ...
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Intuition for Levi-Civita connection?

Levi-Civita connection is usually defined as the unique connection which is torsion free and preserves metric. Question Is there some intuitively transparent constructive way to define it (or ...
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If all balls around two points are isometric… — manifold version

This question is a natural follow-up of this other question, asked earlier today by wspin. Let's say that a metric space $(X,d)$ has two poles if: there are two distinct points $x$, $y$ such that ...
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1answer
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elliptic operators corresponds to non vanishing vector fields

Let $X$ be a non vanishing vector field on a compact manifold $M$. The only differential operator associated with $X$ which I am aware of, is the derivational operator $D(g)=X.g$. Unfortunately ...
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The “Rolle theorem” for sections of a vector bundle

1)Assume that $E\to M$ is a smooth real vector bundle and $\nabla$ is a connection. (We do not assume any metric compatibility since we do not fix a metric on $E$). Assume that ...
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Ellipses on spheres (and other surfaces)

Define an ellipse $E$ on a sphere as the locus of points whose sum of shortest geodesic distances to two foci $p_1$ and $p_2$ is a constant $d$. There are conditions on $\{ p_1, p_2, d \}$ for this ...