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.
3,083 questions
2
votes
2
answers
70
views
extension of the projectivized gradient of a harmonic function
Let $(M,g)$ be a riemannian manifold, $\Delta$ the associated Laplacian, and $\{ f_i \}$ the real-valued eigenfunctions of $\Delta$. Then, $\nabla f_i \in \Gamma ^{\infty } (\mathrm{T} M) $ is defined ...
- 167
4
votes
1
answer
1k
views
Flat Riemannian manifold
Is it true that a Riemannian manifold is flat, if and only if a coordinate transformation $f$ exists, such that the geodesics after transformation is in linear form $\mathbf{y}_t=\mathbf{a}t+\mathbf{y}...
- 8,086
3
votes
4
answers
1k
views
Intrinsic definition of arc length [closed]
Is there an intrinsic way of defining the arc length of a curve in $\mathbb{R}^{3}$, that is without resorting to a parametrization of the curve?
- 3,738
18
votes
2
answers
4k
views
Where is the exponential map a diffeomorphism?
Let $M$ be a closed compact Riemannian manifold.
The exponential map $\mathrm{exp}:TM\to M\times M$ takes $(p,v)$ to $(p,\gamma_v(1))$, where $\gamma_v$ is the geodesic flow at $p$ in the direction ...
- 54.6k
2
votes
1
answer
420
views
Nowhere vanishing, normalized vector field with bounded derivatives
It is well-known that any non-compact manifold admits a nowhere vanishing vector field. If we have a Riemannian metric we may pick such a vector field and normalize it so that at every point it has ...
CommunityBot
- 1
3
votes
1
answer
398
views
Does convex hypersurface necessarily bound a convex domain?
Let $H\in M$ be a convex hypersurface, where $M$ is a complete Riemannian manifold and $H$ is an embedded (complete as a induced metric space) hyper surface without boundary and with positive definite ...
- 8,086
12
votes
0
answers
354
views
A variation on the local Günther inequality
This question is about a variation on the Günther (also known as Günther-Bishop) inequality for manifolds of sectional curvature bounded from above. With Greg Kuperberg, we would deduce from it a ...
- 14.4k
1
vote
0
answers
211
views
Riemann curvature of $S^1$-principal bundle
Let $(M,g)$ be a Riemannian manifold and $\pi:P \to M$ be $S^1$- principal fiber bundle endowed with a connection $\Gamma$. For every $p\in P$ we have,
$$T_pP \simeq T_pV\oplus\Gamma_p$$
Where $V$ ...
CommunityBot
- 1
1
vote
3
answers
535
views
Isometric imbedding of finite metric space into standards spaces [duplicate]
Is it true that any metric space consisting of $n$ points can be isometrically imbedded into $n-1$ dimensional Euclidean space? Hyperbolic space?
(For $n=3$ this is true.) If not, what are necessary/...
CommunityBot
- 1
1
vote
0
answers
125
views
Space with $Ric \geq -(n-1)$
Note that hyperbolic space $H$ has $Ric=-(n-1)$. I want to know :
Question : Does there exists a simply connected open complete Riemannian manifold $M$ s.t.
(1) $ Ric\geq -(n-1)$ on $M$
(2) $ ...
- 1,100
2
votes
1
answer
331
views
What kinds of manifolds admit concave boundary?
We can find many examples of smooth Riemannian manifolds with boundaries whose boundaries are convex. But it seems to me I know no any example of smooth Riemannian manifold with concave boundary. So ...
CommunityBot
- 1
5
votes
1
answer
647
views
Stochastic interpretation of heat kernel on fiber bundle
I'm looking for a stochastic interpretation of the heat equation for vector valued function.
The classical set up is the following :
If $(M,g)$ is a riemannian manifold then we could consider the ...
- 9,853
4
votes
1
answer
468
views
Equivariant Harmonic Maps to R-tree and Korevaar-Schoen Convergence
Thank you for spending time on the following question.
I am trying to make an explicit example of Korevaar-Schoen convergence. The problem I am facing is that I cannot find the limit of the harmonic ...
- 703
0
votes
1
answer
151
views
Area of a plane surface that gives a lot of theoretical problems
Let $\mathbf{r}:(a,b)\times (0,1)\to\mathbb{R}^2\subseteq\mathbb{R}^3$ be a injective application, given by:
$$\mathbf{r}(u,v)=A(u)+v\cdot (B(u)-A(u)), \forall\ (u,v)\in (a,b)\times (0,1)$$
where $A,...
- 18.7k
2
votes
0
answers
168
views
Sources on evolution of submanifolds subject to Ricci flow
I am seeking any textbook or paper addressing the evolution of submanifolds of a manifold undergoing Ricci Flow. Please, any pointer towards this topic is more than welcome.
This old MO post may be ...
CommunityBot
- 1
9
votes
1
answer
1k
views
How submanifolds evolve under Ricci flow?
This may be very naive, since I just started trying to learn Ricci flow; but I couldn't really find any answer after looking for a while in all the textbooks and lecture notes I found online...
...
- 163
1
vote
1
answer
463
views
Singularities in minimal surfaces [closed]
There is a theorem by Jean Taylor that says that an almost minimal set in $\mathbb{R}^3$ can be locally parametrize by the only three possible minimal cones in $\mathbb{R}^3$, the plane, an $Y$ times ...
- 215
6
votes
1
answer
393
views
An unusual metric reconstruction problem
$\newcommand{\bR}{\mathbb{R}}\newcommand{\pa}{\partial}$This questions has some nebulous roots in Morse theory. The most general version goes as follows. Fix an integer $n\geq 2$. Suppose that we have ...
- 34.7k
8
votes
2
answers
224
views
Negatively curved metrics minimizing the length of a homotopy class of simple closed curves
Good afternoon everyone !
I have the following question of Riemannian geometry :
Let $M$ be a smooth closed orientable manifold of dimension at least $3$, and let $\mathcal{T} = \{ $ smooth ...
- 68.9k
2
votes
1
answer
323
views
Control of the metric in isothermal coordinates
Suppose you have a riemannian surface $(\Sigma,g)$, and an open simply-connected set $U \subset \Sigma$. You know that you can find isothermal coordinates - that is a map $\varphi : U \rightarrow D$ ($...
CommunityBot
- 1
8
votes
1
answer
1k
views
Spectrum of the Laplacian on p-forms on the sphere
In this paper the authors give an explicit description of the eigenforms and spectrum of the Laplacian acting on $p$-forms on the round sphere $S^n$, apparently citing an unpublished computation of ...
- 2,446
3
votes
0
answers
109
views
Two questions on topological and geometric structure of projections in a simple $C^{*}$ algebra
Let $A$ be a simple $C^{*}$ algebra. Assume that the space of projections has a connected component homeomorphic to the complex Grassmanian $G(k,n)$. Is it true to say that, for all $k'<k$, the ...
- 356
1
vote
1
answer
183
views
Alexandrov spaces of constant curvature
Let $X$ be locally compact Alexandrov space whose curvature satisfies both inequalities $\geq K$ and $\leq K$. What can be said about such a space? Is it locally isometric to the standard Riemannian ...
- 66
19
votes
4
answers
3k
views
Equations satisfied by the Riemann curvature tensor
It is well known that the Riemann curvature tensor of a metric satisfies
\begin{eqnarray}
R_{jikl}=-R_{ijkl}=R_{ijlk},(1)\\
R_{klij}=R_{ijkl},(2)\\
R_{i[jkl]}=0 \mbox{(1st Bianchi identity)}.(3)
\end{...
- 27.5k
1
vote
1
answer
155
views
Existence of shortest paths in complete Alexandrov spaces
Let $X$ be complete finite dimensional Alexandrov space with curvature bounded from below. Is it true that any two points can be connected by a shortest path? If this is not true in general, it it ...
- 763
10
votes
3
answers
834
views
Rigorous justification that overdetermined systems do not have a solution
There is the following well known and very useful heuristic principle: Assume one has a natural map from the space of $k$-tuples of functions in $n$ variables into the space of $K$-tuples of functions ...
CommunityBot
- 1
8
votes
1
answer
289
views
Closed geodesics in free smooth loop space?
I know very little about these subjects, so I apologise if this is a naive line of inquiry:
Let $M$ be a smooth $n$-dimensional Riemannian manifold. I understand that it is possible to construct an ...
- 25.3k
2
votes
0
answers
812
views
Products between metrics in a product of manifolds
In the "Einstein Manifold" book written by Arthur Besse, chapter 16, there is a notation of a manifold composed by the Cartesian product between two others:
$(M_1\times M_2, f^p(g_1 \times g_2))$
...
- 690
19
votes
2
answers
7k
views
*The* open problem in General Relativity?
Q. Is there a single, clear mathematical question that has emerged as
the open problem in General Relativity?
I ask this on the ~100th anniversary of Einstein's (4-page!) 1915 paper,
"Die ...
CommunityBot
- 1
8
votes
3
answers
750
views
Classification of natural invariants of Riemannian structures
Before I formulate my question, let me remind P. B. Gilkey's characterization of Pontryagin forms,following the paper "On the heat equation and the index theorem" by Atiyah, Bott, Patodi.
By ...
- 21.5k
2
votes
1
answer
232
views
Shortest paths in Alexandrov spaces
Let $X$ be an Alexandrov space with curvature bounded from below (if necessary, $X$ might be assumed to be finite dimensional or even compact).
Question 1. Is it true that every point of $X$ has a ...
- 763
3
votes
0
answers
109
views
Lower boundedness of the Ricci curvature [closed]
I know that this is going to be slightly vague, but it's itching me: why is lower boundedness of the Ricci curvature required by some of the very important theorems in Riemannian geometry? The ...
- 5,407
7
votes
0
answers
352
views
Finitely generated projective modules over the algebra of sections of the Clifford bundle
Consider a (pseudo-)Riemannian manifold $(M,g)$ and the corresponding Clifford bundle $Cl_g(T^*M)$. Let $R$ be the algebra of sections of $CL_g(T^*M)$, with point-wise multiplication. What are the ...
- 21.5k
3
votes
1
answer
141
views
Conditions for tubular hypersurfaces to be a Riemannian product
Let $(M,g)$ be a Riemannian manifold of dimension $n$ and $P$ a submanifold of dimension $k.$ Let us define the tube of radius $r$ about $P$ by
$$T(P,r):=\{x\in M: d(x,P)\le r\}$$ and the tubular ...
- 763
2
votes
1
answer
244
views
Symplectic reduction: from indefinite signature to Riemannian signature
Let $(M,g,\omega)$ be a $d$-dimensional manifold equipped with a metric $g$ of signature $(t,s)$, $d = t+s$, and a symplectic form $\omega$. Let us assume that a Lie group $G\subset Isometries(M,g)$ ...
- 4,787
3
votes
1
answer
223
views
structure of metrics on a compact manifold
is there a reference on the structure of the space of metrics on a compact manifold that induce a given measure $\mu $?
i have a given manifold $M$, a given measure $\mu$ with an everywhere positive $...
- 167
1
vote
0
answers
165
views
What is the intersection of Spin(7) and U(4)?
I'm just curious from Berger's classification of Riemannian holonomy, how do Spin(7) manifolds intersect the other types of Riemannian manifolds?
In particular, what is the intersection of Spin(7) ...
- 1,347
2
votes
0
answers
148
views
Derivation of gradient of SSE in Geodesic Regression
On page 79 (or page 5) of this this paper the gradient of the SSE of the Geodesic model is described explicitly. My question is how are these equitations derived in detail; where can I find the ...
CommunityBot
- 1
2
votes
1
answer
358
views
Perimeter of ellipse: Combination of two geometries
Is there a Riemannian metric $g$ on $\mathbb{R}^{2}$ such that for every ellipse $\gamma$ in the plane we have:$$\text{The Euclidien perimeter of}\; \gamma=\lambda (g\text{-diameter of}\;\gamma)$$...
- 4,787
2
votes
1
answer
271
views
References on the Free Loop Space
I intend to approach the paper of Wolfgang Ziller: "The Free Loop Space of Globally Symmetric Spaces", but I need the proper background on the foundations of the study of Free Loop Spaces. I obtained ...
- 16.5k
0
votes
0
answers
121
views
Positive curvature of the boundary away from a point implies regularity?
In a paper I'm refereeing, the authors make use of the following geometric fact:
Let $U$ be an open subset of $\mathbb{R}^2$. If there is a point $p\in \partial U$ so that $\partial U \backslash p$ ...
- 6,210
1
vote
1
answer
250
views
Global geometry measures for Riemannian manifolds
I'm working on a stochastic algorithm and considering it to apply in case of any curved space (manifolds). But in order to make the algorithm as efficient as possible I want to include in it some ...
CommunityBot
- 1
3
votes
0
answers
615
views
Estimates of eigenvalues of elliptic operators on compact manifolds
The classical Weyl law says that if $\Delta$ is the Laplace operator on functions on a compact Riemannian manifold $(M^n,g)$, $n>2$, then its $k$th eigenvalue satisfies the asymptotic formula
$$\...
- 21.8k
2
votes
1
answer
346
views
Closed geodesics that cross one another frequently
Let $S$ be a smooth, closed, genus zero surface in $\mathbb{R}^3$.
$S$ has at least three simple (non-self-intersecting), closed geodesics by
a theorem of Lyusternik and Shnirel'man.
Alternatively, ...
- 108k
3
votes
1
answer
659
views
Short time existence on Hyperbolic Ricci flow in non-compact case
We know
Laplace equation (elliptic equations)
$ Δ u = 0$
Heat equation (parabolic equations)
$u_t − Δu = 0$
Wave equation (hyperbolic equations)
$u_{tt} − Δu = 0$
we have
- Hyperbolic geometric ...
- 39.1k
0
votes
1
answer
103
views
Lie group action on a finite dimensional flat manifold
Consider a finite dimensional flat Riemannian manifold $M$ quotiented by an action of a finite dimensional Lie group $G$, giving rise to the quotient $Q$.
First, assume that the action is isometric. ...
- 14.4k
5
votes
3
answers
1k
views
Area of metric spheres in Riemannian manifolds
I am trying to estimate the integral $\int \mathbb{e} ^{-d(x_0,x)^2} \mathbb{d}x$ on a Riemann manifold $(M,g)$, for some arbitrary fixed $x_0 \in M$ and $d$ the usual distance. The only thing that I ...
- 34.7k
3
votes
1
answer
249
views
length comparison on negatively curved surfaces
Suppose $g_1$, and $g_2$ are two Riemannian metrics on a closed surface $S$, provided that the Gaussian curvature $K_{g_1}$ $<$ $K_{g_2}\leq -1$. Denote by $\mathcal{C}$ the set of free homotopy ...
- 165
7
votes
1
answer
898
views
Sharp Gaussian upper bounds on Heat Kernel
I am looking for references (with proof) for the following statement:
Let $(M, g)$ be a Riemannian manifold with bounded curvature and let $p_t(x , y)$ be the heat kernel of $M$. Let $K$ be compact ...
- 5,407
4
votes
0
answers
324
views
How to check if a manifold can be foliated by strictly convex hypersurfaces?
Let $M$ be a compact Riemannian manifold with boundary.
How can one recognize whether the manifold can be foliated by strictly convex hypersurfaces?
An exact definition is given below.
If the ...
- 8,086