# 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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### Local Sobolev embedding on complete Riemannian manifold

Let $(M,g)$ be a complete Riemannian $m$-manifold, with bounded geometry and $m\geq2$. Suppose $Ric\geq(n-1)\kappa$. Let $B_p(r)$ be a geodesic open ball.
Q Can we find a constant $C=C(\kappa,r,m)$(...

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### volume of parallelotope in $L^2(\mathbb R).$

Let $L^2(\mathbb R)$ is complex Hilbert space with standard inner product.
Does it make sense to talk of volume of parallelotope formed by following vectors in $L^2(\mathbb R):$ say, e.g.,
$$\{ f(...

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### On the infimium of a functional

Let $(M^n,g)$ be a closed Riemannian manifold. Define
$$\lambda(g)=\inf\{\mathcal{F}(g,f),\;0<f\in C^{\infty}(M),\; \int_Mf^2\;d\nu=1\},$$
where $$\mathcal{F}(g,f)=\int_M\left(|\nabla f|^2+ af^2\...

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### Intrinsic Reach for a Riemannian manifold

The reach of a set $X\subseteq \mathbb{R}^d$ is the supremum of all $r \geq 0$ such that for all $y\in X^c$ with $dist(y,X)<r$ there is a unique $x\in X$ with $dist(y,x)= dist(y,X)$.
My question: ...

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### Eigenvalues of geometric operators along geometric flows

I have two questions:
1- what is the relation between eigenvalues of geometric operators such as Laplace operator and topology or geometry of a Riemannian manifold?(please give an example if possible)...

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### Critical metric for an Hilbert action?

Suppose that $\omega$ is an 1-form on a Reimannain Manifold $(M,g)$ and $s$ is a $(0,2)$ symmetric tensor which be considered as $(1,1)$ symmetric tensor whenever it is convenient. If for all $s$ the ...

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### Geodesic connectedness in static Lorentz manifold vs connectedness by trajectories with potential in Riemann manifold

What is the relationship between the study of geodesic connectedness in a standard static Lorentz manifold and the connectedness of two points by trajectories with potential (i.e. solutions to $x''(t) ...

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### Quantitative upper bound on mean curvature of an isometric embedding

By Nash embedding theorem, any complete Riemannian manifold $M$ can be isometrically embedded in $\mathbb{R}^N$, for sufficiently large $N$.
The proof of the theorem is quite involved, and it is not ...

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### regularity of harmonic forms on manifolds-with-boundaries

Let $M$ be a smooth, bounded, oriented Riemannian manifold-with-boundary. Let $\alpha$ be a harmonic differential $p$-form on $M$, subject to the boundary condition $\alpha\wedge\nu^\sharp|\partial M =...

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### Surjectivity of Pseudo-Riemannian exponential map on geodesically complete manifolds

Suppose one has a geodesically complete pseudo-Riemannian manifold $M$ i.e. the exponential map is defined for all tangent vectors on the manifold. Can one make a sensible statement about whether (or ...

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### Complex Riemannian metrics over real manifolds

There is a huge literature on complex manifolds and natural metrics over them, but I was unable to find references about Riemannian complex metric on real manifolds (for which we complexify tangent ...

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### Configurations of $n$ points modulo isometries of the ambient space

Let $M$ be a Riemannian manifold and let $n$ a positive integer. Denote by $F_n(M) \subset M^n$ the space of all $n$-tuples of pairwise distinct points from $M$. The isometries of $M$ act co-ordinate ...

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### Geodesics in the almost flat manifolds

Let $(M,g)$ be a complete Riemannian manifold and $p$ a fixed point on $M$. Assume that the curvature is bounded by $\Lambda^2$ on $B(q,10)$ such that $\Lambda$ is a small number, we consider the ball ...

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### convergence and a mean curvature condition imply convexity

I have a question regarding the proof of theorem 4.6 in https://arxiv.org/abs/1007.3899 (Hall's conjecture).
Let $S^2$ be the class of Borel subsets in $\mathbb{R}^2$ with finite and positive ...

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### Can a harmonic function on a topological cylinder have critical points?

Let $M$ be an oriented closed smooth manifold, and let $C=M\times[0,1]$, the cylinder over $M$. Let $g$ be an arbitrary Riemannian metric on $C$ (in particular, $g$ may look nothing like a product ...

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### Gradient estimate for Poisson equation on manifold

In Gilbarg-Trudinger's book 'Elliptic Partial Differential Equations of Second order', the maximum principle is used to derive the following gradient estimates for Poisson equations on Euclidean ...

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### Is there a Riemannian submersion from $Gl(2,\mathbb{R})$ to the Poincare half plane?

Let $\mathbb{H}$ be the Poincare half plane with the hyperbolic metric. Let $Gl(2,\mathbb{R})$ be equipped with a left invariant metric?
Is there a Riemannian submersion from $Gl(2,\mathbb{R})$ to ...

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### Control distance of the boundary

Let $(M,g)$ be a Riemannian manifold with boundary $\partial M$. Let $(g_t)_{0\leq t\leq T}$ be a family of Riemannian metrics with $g_0=g$. Suppose that $\partial M$ has at least two connected ...

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### What is wrong with the derivation?

Let $(M^n,g)$ be a Riemannian manifold, and $T$ a symmetric $(1,1)$-tensor field, i.e., $\langle T(X),Y\rangle = \langle X,T(Y)\rangle $. For convenience, denote $$\Delta_Tu=\sum_i\langle \nabla_{e_i}\...

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### If the total space of circle bundle over hyperbolic manifold admits Riemannain metric of non-positive sectional curvature?

If the total space of circle bundle over higher genus surface admit Riemannian metric of non-positive sectional curvature?
I wish to use the result about the question and find Leeb's work 3-...

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### Flat solvmanifolds?

I was looking for some reference on solvmanifolds and came up with a paper by A. Morgan tilted "The classification of flat solvmanifolds". I know there is a complete classification of flat manifolds ...

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### Sheaves on solenoids

Let $(X_n)$ be a tower of finite covering maps of compact smooth manifolds, with $f_{s,t} : X_t\to X_s$ the maps, and $\Lambda_n := f_{n,0}^{-1}\Lambda$, with $\Lambda$ the constant abelian sheaf on $...

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### Do we have estimate like $\int_\gamma \alpha \le |\alpha| \cdot |\gamma|$?

Let $(X,g)$ be a compact smooth Riemannian manifold. It is known that $H^1(X, \mathbb R)\cong \mathrm{Hom} (\pi_1(X), \mathbb R)$, namely there is a natural pairing
$$
H^1(X) \times \pi_1(X) \to \...

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### An integral estimate in conformal geometry

Let $g_0$ be a Riemannian metric of positive scalar curvature $R_0$ on a closed 4-manifold, and for some fixed constant $C$, define the set
\begin{equation}
\mathcal{S} = \{u\in C^\infty(M): ||u||_{W^{...

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### Differentiating Riemannian logarithmic map

Let $(M,g)$ be a Riemannian manifold, geodesically complete, and assume logarithms are well defined and smooth.
Let $c: I\to M $ be a smooth path in $M$, and $x\in M$. Can we say something about $$\...

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### Minimizing geodesics in incomplete Riemannian manifolds

Let $(M, g)$ be a Riemannian manifold, not necessarily complete. Let $x$ be a point in $M$, and let $r>0$ be such that the exponential map $\operatorname{exp}_x$ is defined on an open ball $B=B(0,r)...

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### Conformally flat homogeneous spaces

Let's say we have a homogeneous space $H\backslash G$.
Is it possible to tell whether this homogeneous space admits a conformally flat metric just from its group structure?
I am particularly ...

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### Diffusion generators with gradient vector fields

Let $\mathcal{A}$ be a second order operator on an $n$-dimensional smooth manifold $M$, expressed in Hörmander form as
$$\mathcal{A}=X_0+\frac{1}{2}\sum_i^kX_i^2,$$
where $X_0,X_1,...,X_k$ are ...

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### Elliptic regularity on compact manifold without boundary

Let $(M,g)$ be a Riemannian compact manifold without boundary, and $\Delta$ is the Laplace-Beltrami operator on $M$. Is there any result on the elliptic regularity like this:
For any $u\in H^1(M)$, ...

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### Is there Brownian motion on Alexandrov spaces?

It is well known that there is a notion of Brownian motion on smooth Riemannian manifolds.
I am wondering if there is a more general notion of Brownian motion on finite dimensional Alexandrov ...

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### Simple application of Bochner--Reilly--Weitzenböck type formulas

I am looking for simple (but not worn-out) application of Bochner--Weitzenböck type formulas in comparison geometry. (I want to use it as a motivation for students.)
The vanishing theorems and ...

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### Existence of a certain kind of compact spin manifold with boundary

For a compact spin Riemannian manifold $(M^n,g)$ without boundary, $n \not\equiv 3\mod 4$, it is well-known that the Dirac operator associated with a fixed spin structure $S\rightarrow M$ has real, ...

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### Vector-valued forms in Riemannian geometry

Suppose $(M,g)$ is a Riemannian manifold. I want to find a vector-valued $2-$form
$T$ such that, for any vector fields $X,Y,Z$ on $M$,
$$
g(T(X,Y),Z)=g(T(Z,X),Y)\,.
$$
As a motivation, consider the ...

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### The developing map of conformally flat manifold

There is one sentence I don't understand in some paper.
"A simply connected and conformally flat three mainifold can be conformally immersed into $S^3$" by the means of a developing map.
Is any ...

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### Symplectic connections are (locally) Levi-Civita connections

I was wondering... Is every symplectic connection $\nabla$
on some symplectic manifold $(M,ω)$ the Levi-Civita connection of some metric $g$ on $M$? What about the local statement?

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### Sectional curvature of leaves of foliation

Given a $k$- dimensional foliation $F$ of a riemannian $n$-manifold $M$, with the property that the leaves of the foliation have constant sectional curvature $s$, for some $s$, is it true that $M$ ...

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### Deforming metrics from non-negative to positive Ricci curvature

Given a closed Riemannian manifold $(M,g)$ with non-negative Ricci curvature and $dim\geq 3$, when can we deform the metric to a positive Ricci curved one?
I know it's impossible in general due to ...

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### Smooth functions on sphere

Let $u$ be a smooth function defined on the unit sphere $S^2$. Assume $u$ has two local maxima, two local minima, and two saddle points (a total of 6 critical points). Does there exist a plane $P$ ...

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### Killing vector fields of a conformally flat Riemannian metric

Let $f: \mathbb{R}^n \to \mathbb{R}$ be a smooth function and let's consider the conformally flat Riemannian metric $g = e^f \delta_{ij} dx^idx^j$ on $\mathbb{R}^n$.
Is it true that the Killing ...

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### Trace free Codazzi Tensor on Hyperbolic manifolds

Does there exist trace free nontrivial symmetric Codazzi Tensor on closed manifold with constant sectional curvature -1?
I know, locally all Codazzi Tensors on closed manifold $(M, g)$ with constant ...

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### An upper bound for the number of singularities of a transversal vector field isometric to the zero field

Let $(M,g)$ be a Riemannian manifold. We equip the tangent bundle $TM$ with the Sasaki metric $g_s$.
A smooth vector field $X:M \to TM$ is called a transversal vector field if $X(M)$ is transverse ...

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### Positive Ricci curvature on fiber bundles

My advisor and I are working on Ricci curvature and an anonymous referee pointed out the following conjecture:
Let $F\hookrightarrow M\stackrel{\pi}{\to}B$ be a fiber bundle from a compact manifold ...

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### Eigenfunctions of square root of Laplacian in an arbitrary Riemannian manifold?

Please forgive me for my inability to pose a mathematics question properly.
In one dimension the eigenfunctions of Laplacian (simply double derivative) are also eigenfunctions of its square root (...

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### The converse to the positive mass theorem

Let $(M^n,g)$ be an asymptotically flat manifold of decaying-order $\tau>\frac{n-2}{2}$, the positive mass theorem states that if the scalar curvature $S_g$ is non-negative, then the ADM mass $m_g$...

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### Einstein warped product manifold Ricci flat

Let $(M,g)=(N,\ddot{g})\times f(B,\bar{g})$ be an Einstein warped-product manifold Ricci flat (i.e. $Ric=\lambda g$ with $\lambda=0$) where $f:N \rightarrow (0, \infty)$ (positive scalar function) and ...

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### Generalization of Bieberbach's second theorem

Let $F_0$ and $F_1$ be compact flat manifolds of dimensions $k$ and $m$, respectively, where $k \geq m$. Suppose $f : \pi_1(F_0) \to \pi_1(F_1)$ is a surjective homomorphism. Consider the covering ...

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### Exponential decay of resolvent kernel

For the integral kernel of the Laplacian $\Delta$ on $\mathbb{R}^n$, consider the resolvent $R(\lambda) := (\lambda - \Delta)^{-1}$ and let $R(\lambda; x, y)$ be its kernel, which is a smooth function ...

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### Orthonormal vector fields on a Riemannian manifold

Let $(M,g)$ be a Riemannian manifold. We equip the tangent bundle $TM$ with the Sasaki metric $g_s$.
Assume that $X: M \to TM$ is a vector field on $M$.
We say that $X$ is an orthonormal vector field ...

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### Lichnerowicz-Obata theorem for symmetric 2-tensor

Let $(S^n/\Gamma,g)$ be the standard space-form with constant sectional curvature, where $\Gamma \subset O(n+1)$ is a finite group. If there exists a nonzero transverse-traceless symmetric 2-tensor $h$...

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### The Hausdorff dimension of the union of singular orbits and exceptional orbits

Suppose we have a compact connected Lie group $G$ acting as isometries on a compact manifold $M^n.$ Then is it necessarily true that the Hausdorff dimension of the union of singular and exceptional ...