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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Volume growth and visibility in hyperbolic spaces
I am interested in the following two questions for complex, quaternionic and octonionic hyperbolic spaces, equipped with their usual metrics and measures:
For brevity, I will denote the volume of the ...
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Can we convert any non-vanishing vector field into geodesic field by changing metric?
If $M$ is a smooth closed manifold together with a non-vanishing (maybe unit) vector field $X$. In what condition can we construct a Riemannian metric on $M$ s.t $X$ be the geodesic field of on $TM$?
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Is the Lie derivative of a harmonic form also a harmonic form?
On Helgason's book "Differential Geometry, Lie Groups, and Symmetric Spaces" it is said that the Lie derivative along a left-invariant vector field of an harmonic form is again a harmonic form. This ...
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Solutions of this system of PDE's
This question is related to the existence of Einstein metrics on tangent bundles where the metric is induced by the isotropic almost complex structures on the tangent bundle. I'm trying this on the ...
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Calabi - Yau Manifolds
I just started reading about Calabi-Yau manifolds and most of the sources I came across defined Calabi-Yau manifold in a different way. I can see that some of them are just same and I can derive one ...
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Bubbling example for harmonic maps
Let us consider, a closed Riemannian surface $(\Sigma,h)$ and a compact Riemannian manifold $(N,g)$ with dimension greater than $3$. If we are given a sequence of harmonic maps $u_n:(\Sigma,h) \...
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Self-contained book on Ricci Flow/Geometric Analysis
Can someone please tell me whether there is any self-contained book on Geometric Analysis/Ricci Flow/analytic techniques used in Riemannian Geometry? By self-contained I mean it does not assume that ...
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Ricci flow preserves holonomy
Could someone please give me a reference where I can find a complete proof of the result Ricci flow preserves holonomy? Is there any way to prove that Ricci flow preserves Kahler condition without ...
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On generalized Tanaka connection
Many authors used the Tanaka connection in their papers such as
[1]
to define new Tanaka connection so-called Generalized Tanaka connection $^*\nabla$ on a contact Riemannian manifold $(M,\eta,\xi,\...
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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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Curvature and Parallel Transport
Here is an updated formulation of the question, which is more precise and I think completely correct:
Suppose $M$ is a Riemannian manifold. Pick a point $p$ in $M$ and let $U$ be a neighborhood of ...
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Does uniform convergence of (Riemannian) distances implies convergence of conformal structures?
I don't know much about the Teichmüller space, so maybe the question I ask is well known; still I can not find the answer by myself...
Let $\Sigma$ be a closed surface. Let $g_m$ be a sequence of (...
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Estimates for radii of slices for proper Lie group actions
Let $G$ be a Lie group acting properly on a smooth manifold $M$, and equip $M$ with a Riemannian manifold that is adapted to the foliation by orbits. The celebrated theorem of Palais is that there ...
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Gauss' theorem for null boundaries
Prenote: I have asked this question first on math stackexhange, but a user suggested that mathoverflow might be a better place for this question. Upon thinking about it I have agreed with him and copy-...
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Isometric and conformally equivalent surfaces in $\mathbb R^d$ with the same boundary
Let $X$ be a compact smooth surface in $\mathbb R^d$ with smooth boundary $\partial X$. Is it possible to find another smooth surface $Y$ in $\mathbb R^d$ such that $\partial X = \partial Y$ and $Y$ ...
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Limit space of a sequence of Riemannian manifolds with uniformly bounded below Ricci curvature
Let $\{M^n_i\}_{i=1}^\infty$ be a sequence of closed smooth Riemannian $n$-dimensional manifolds with uniformly bounded below Ricci curvature and uniformly bounded above diameter. The Gromov ...
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Geometric interpretation of horizontal and vertical lift of vector field
In many References such as D.E. Blair, Riemannian Geometry of Contact and Symplectic Manifolds chapter 9, and Differential Geometric Structures
By Walter A. Poor Page 54; the horizontal and vertical ...
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Nice applications of Liouville's theorem
I need a big list of nice-looking and simple applications of Liouville's theorem on geodesic flow in Riemannian geometry.
Please help.
Examples:
A Riemannian manifold with finite volume does not ...
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The momentum constraints in the ADM formulation of general relativity
Suppose that the space-time has a time function. Let $g_{ij}$ be
the Riemannian metrics of the time slices, and $K_{ij}$ be the second
fundamental forms. It is by Codazzi equation that
$$
D^{i}(K_{ij}...
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The midpoint geodesic
Let $(M,g)$ be a complete simply connected Riemannian manifold with non-positive curvature. Because of the Hopf-Rinow theorem, any two points are connected by a geodesic segment.
Pick three distinct ...
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The Lie algebra of Harmonic functions
Is there a Riemannian metric $g$ on $\mathbb{R}^{2}$ with corresponding volume form $\omega= \sqrt{det(g_{ij})} dx \wedge dy$ and the corresponding Laplace operator $\Delta$ such that the space ...
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Umbilic points on Euclidean hypersurfaces
Every smooth embedding of $S^2$ into $\mathbb{R}^3$ has at least one umbilic point (in fact, the recent proof of the Caratheodory conjecture yields two such points). The usual proof of this is to use ...
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Twisting Spinor Bundles with Line Bundles
In a paper I am reading, the following framework was given: Let $S$ be a spinor bundle, over a Riemannian manifold $M$, with Clifford action
$$
c:S \otimes \Omega^1(M) \to S.
$$
Moreover, let $E$ be ...
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Chebyshev net in 3D
I would like to know the reasons why the existance of Chebyshev net in 3D-case is problematic.
This question boils down to the PDE described below.
(I do not know much about PDEs, so feel free to say ...
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Does every smooth manifold admit a metric with bounded geometry and uniform growth?
Let $M$ be a smooth manifold, $g_M$ a Riemannian metric, and consider for $x\in M$ the volume growth function, $gr_x$ that maps $r>0$ to the volume $vol_{g_M}(B(x,r))$. My interest is to see ...
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Does every Zoll metric on $\mathbb{S}^2$ arise from a perturbation of the round metric?
The introduction here states 'A formal perturbation argument
of Funk later indicated that, modulo isometries and rescalings, the general Zoll
metric on $\mathbb{S}^2$ depends on one odd function $f:\...
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Comparison of angles in Alexandrov space
Let $X$ be a finite dimensional Alexandrov space with curvature bounded below. Let $p\in X$ be a fixed point.
Is it true that for any $\varepsilon >0$ there exists $\delta>0$ such that for any $...
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$L^2$-estimate for an elliptic equation
Given $(M, g)$ a compact Riemannian manifold of dimension $n \geq 3$, we consider the following equation for the function $\phi$:
$$
-\Delta \phi + R \phi + \tau^2 \phi^{N-1} = \frac{A^2}{\phi^{N+1}}
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First eigenvalue for strictly convex domains
Let $M^n$ be a compact Riemannian manifold with boundary, suppose 1). $Ric(M)\ge (n-1)$ and 2). the principle curvatures of the boundary is bounded from below by $h\ge 0$. Is there any results on the ...
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Can a Riemannian metric be analytic in non-analytically different coordinates?
Suppose I have two coordinates on the same (subset of a) Riemannian manifold.
If the metric tensor is analytic in both coordinates, is the change of variables between them necessarily analytic?
In ...
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Decomposition of pullback metric
Let $(M^3,g)$ be a complete riemannian manifold and $\Sigma ^2\subset M^3$ a embedded minimal compact surface. Consider the immersion $\phi: \Sigma \times [0,\varepsilon)\to M$ given by
$$\phi(p,t)=\...
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Question on Weil-Petersson metric on Teichmuller space
I'm reading Ahlfors' original articles about Weil-Petersson metric: "Some remarks on Teichmüller's space of Riemann surfaces" and "Curvature properties of Teichmüller's space".
The tangent space at ...
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Why is it easy to compute the first and fourth moments for random chord length in a convex solid?
Recently I was led to some considerations in geometric probability, a field pretty far from any specialization of mine. (Context: I was working with a collaborator on a question about mean escape ...
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A connection on $Hom( E,E)$ whose parallel transport is compatible to parallel transport of $E$
According to the answer of Sebastan and previous edit of Ben McKay I revise my post as follows:
Assume that $E$ is a vector bundle over a manifold $M$ with a connection $\nabla$.
Is there a (...
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Recover Embedding from Metric
Suppose that $M$ is an embedded sub-manifold of $D$-dimensional Euclidean space $E^D$, with embedding $\phi:M \hookrightarrow E^D$; the embedding is not explicitly known.
And suppose that I know ...
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Isometric embedding of SO(3) into an euclidean space
Consider $SO(3)$ with its bi-invariant metric and $R^n$ the euclidean space of dimension $n$. What is the minimal value of $n$ such that there exists an isometric embedding $f: SO(3) \to R^n$?
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Is a G-invariant metric always Kähler-Einstein?
Suppose there is a Hermitian symmetric space of compact type $X$. It is realized in the following way: $X\hookrightarrow\mathbb{P}^N$ and equipped with the induced Fubini-Study metric $g$.
What's ...
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asymptotic behavior of Lipschitz constants of sectional curvature
I'm studying the paper "Measure Concentration and the Topology of Positively-Curved Riemannian Manifolds" (https://arxiv.org/pdf/1402.4947v1.pdf) and I have some problem in understanding the proof of ...
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Reference: Finsler Derivative?
On the wikipedia page "Generalizations of derivative" the author mentions: " in Finsler geometry, one studies spaces which look locally like Banach spaces. Thus one might want a derivative with some ...
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How do spectrums interact with bi-Lipschitz maps?
If it makes things simple, we can just stick to bi-Lipschitz maps from $S^k \rightarrow \mathbb{R}^d$ (w.r.t geodesic distance on the sphere with the standard round metric and the $2-$norm on the ...
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Counting limit cycles via curvature in Riemannian geometry
In this post we would like to give a possible new approach to the second part of the Hilbert 16th problem
First we give a short introduction:
A quadratic system is a polynomial vector field on ...
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The radially symmetric isoperimetric problem
A solution to the $l$-isoperimetric problem on a Riemannian surface $(M,g)$ is a smooth closed curve $\gamma \subset M$ of length $l$ which minimizes the isoperimetric constant:
$$h(\gamma) = \frac{l}{...
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An answer to this system of PDE's
Planning of the question:
Let $(M,g)$ be a Riemannian manifold and $TM$ be its tangent bundle
The isotropic almost complex structures $J_{\delta , \sigma}$ were introduced by Aguilar on the ...
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Euler characteristic, Gauss-Bonnet, and a product formula
I know very little about the Pfaffian or how it works, and I'm new at Riemannian geometry in general. But I was wondering if there is some way to make this "intuitive" argument for the fact that a ...
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Algebraic (semi-) Riemannian geometry ?
I hope these are not to vague questions for MO.
Is there an analog of the concept of a Riemannian metric, in algebraic geometry?
Of course, transporting things literally from the differential ...
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Ricci curvature and killing form
Motivated by this question we ask:
Is there any relation between the Ricci curvature of a Lie group and the killing form of its Lie algebra?Under what conditions, they are proportional to each ...
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sequence of graphs converge in the sense of varifold to multiplicity 2 plane
Say in $R^3$, is there a sequence of smooth graphs $f_n$ over some plane P, such that the graphs as submanifolds in $R^3$ converge in the sense of varifold (as Radon measures on $R^3 \times Gr(2,3)$ ) ...
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Curvature of a Lie group
Since a lie group is a manifold with the structure of a continuous group, then each point of the manifold [Edit: provided we fix a metric, for example an invariant or bi-invariant one] has some scalar ...
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Existence of a brake geodesic on a non compact Riemannian mfd
I am interested how to find a geodesic (if it exists) on a Riemann manifold s.t.
the geodesic connects 2 different points on the edge of the manifold
the metric is positive definite everywhere on the ...
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minimal surfaces in $S^n$
Thanks to Choi-Schoen theorem, we know that the space of embedded minimal surfaces into $S^3$ of fixed genus is compact. My question are simples:
Can we remove the embeddness assumption?
Can we ...
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