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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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81 views

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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73 views

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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77 views

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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83 views

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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39 views

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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31 views

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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28 views

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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114 views

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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1answer
82 views

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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36 views

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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157 views

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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103 views

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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1answer
73 views

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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136 views

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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104 views

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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1answer
130 views

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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1answer
175 views

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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1answer
225 views

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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138 views

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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1answer
197 views

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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1answer
143 views

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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116 views

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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1answer
92 views

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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4answers
331 views

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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1answer
85 views

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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41 views

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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3answers
277 views

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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1answer
86 views

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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1answer
396 views

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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135 views

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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80 views

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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1answer
174 views

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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2answers
305 views

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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3answers
329 views

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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2answers
356 views

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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6answers
2k views

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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1answer
138 views

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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0answers
122 views

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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1answer
234 views

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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116 views

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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2answers
185 views

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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2answers
169 views

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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1answer
99 views

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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0answers
111 views

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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48 views

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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1answer
112 views

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 ...