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.

learn more… | top users | synonyms

75
votes
6answers
2k views

John Nash's Legacy

It would seem that John Nash and his wife Alicia died tragically in a car accident on 5/23/15 (reference). My condolences to his family and friends. Maybe this is an appropriate time to ask a ...
2
votes
0answers
76 views

Spherical cap is the only compact constant mean curvature surface bounded by a circle

I would like to see that the only compact rotationally invariant constant mean curvature surfaces with boundary a planar circle, are either a planar disk or a spherical cap. This is stated in the ...
0
votes
0answers
29 views

Does anyone know how to describe the zero set of the Jacobian of injective harmonic maps in space?

For example consider the following question: Let $\mathbb{B}^m$ be hyperbolic space and let $f : \mathbb{B}^m \rightarrow \mathbb{B}^m$ be harmonic $K$-qc map. Whether $f$ has critical points on ...
0
votes
1answer
79 views

Finding Riemannian metric for this geodesic

In a $d$-dimensional manifold, given a geodesic equation $\gamma^i(t)=a^i\phi(tb^i),i\in 1\ldots d$, where $\phi:\mathbb{R}\rightarrow\mathbb{R}^+$ is an increasing function, $a^i>0,b^i$ are ...
0
votes
0answers
35 views

Estimation of connection ignoring the inverse parallel transport in manifolds open in Euclidean space [on hold]

Let $(M,g)$ be a Riemannian manifold, with parallel transport $P_{t_1,t_2}$ from time $t_1$ to time $t_2$. We know that, along a curve $c$: $$ \nabla_{c} V(t)= lim_{h\to 0} ...
0
votes
1answer
126 views

Riemann isometry vs Euclidean bi-Lipschitz mapping

Assume that $\gamma$ is a rectifiable Jordan curve in the complex plane of length $2\pi$. Then there exists a Riemann isometry $f$ between $\gamma$ and the unit circle $T$. My question is, does this ...
9
votes
0answers
185 views

Vector field built from connection and metric

Consider a smooth finite-dimensional manifold $M$ with metric $g$ and connection $\nabla$. For some local coordinate system, denote by $g^{\alpha \beta}$ the inverse of the metric tensor and by ...
1
vote
0answers
53 views

Can a cylinder be regarded as a Riemannian manifold? [on hold]

Consider the surface of a bounded cylinder consisting of a top,bottom and side part together with the metric induced by the euclidean norm on $\mathbb{R}^3$. Can this space be regarded as a Riemannian ...
2
votes
2answers
46 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 ...
4
votes
1answer
247 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 ...
2
votes
4answers
531 views

Intrinsic definition of arc length [on hold]

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?
6
votes
1answer
1k views

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, ...
6
votes
2answers
207 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 ...
2
votes
1answer
103 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 ...
2
votes
1answer
52 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 ...
11
votes
0answers
182 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 ...
1
vote
0answers
135 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 $V$ ...
1
vote
3answers
159 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 ...
5
votes
0answers
185 views

Intersections of open balls in manifolds

This question is motivated by the post Uncountable intersections of open balls in a separable metric space. The general problem is the following: given a connected Riemannian manifold $M$, what are ...
0
votes
0answers
76 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) $ ...
30
votes
6answers
3k views

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 ...
1
vote
1answer
211 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 ...
0
votes
0answers
119 views

Gromov's defenition of Content of Ball

Let $B(p, R)$ denote the metric ball of radius $R$ centered at $p$ in a manifold. Then Gromov defined the content of the ball by $$Cont(B(p,R))=rank(H_*(B(p, R/5))\to H_*(B(p,R))) $$ and he remark ...
2
votes
1answer
108 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 ...
3
votes
1answer
137 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 ...
0
votes
1answer
117 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 ...
2
votes
0answers
75 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 ...
7
votes
1answer
558 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... ...
15
votes
5answers
2k views

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 ...
1
vote
1answer
125 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 ...
8
votes
3answers
599 views

is there a global obstruction for a diffeomorphism to be an isometry?

Let $V$ be a finite dimensional vector space. Let us call an automorphism $T:V\rightarrow V$ admissible if there exists an inner product $\langle , \rangle$ on $V$ making $T$ an isometry. We know ...
5
votes
1answer
187 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 ...
5
votes
2answers
109 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 ...
2
votes
1answer
77 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 $\varphi : U \rightarrow D$ ...
5
votes
0answers
284 views

Kernel of Bianchi operator: Is a (smooth tame) Frechet manifold?

Let $M$ be a smooth compact manifold, $\mathcal{S}=\Gamma(\odot^2T^*M)$ the smooth tame Frechet space of smooth symmetric $2$-covariant tensors, and $\mathcal{M}=\Gamma(\odot^2_+T^*M)$ the smooth tame ...
5
votes
1answer
190 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 ...
3
votes
0answers
95 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 ...
4
votes
1answer
244 views

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 ...
1
vote
1answer
71 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 ...
1
vote
1answer
174 views

Is the structure constant additive on connected components?

This is the reanimation of a question which already got an answer, that I did not fully understand. Coming back to it, after let it sit in a corner for some time, I keep not getting the point. I would ...
59
votes
4answers
7k views

When can a Connection Induce a Riemannian Metric for which it is the Levi-Civita Connection?

As we all know, for a Riemannian manifold $(M,g)$, there exists a unique torsion free connection $\nabla_g$, the Levi-Civita connection, that is compatible witht metric. I was wondering if one can ...
15
votes
4answers
889 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) ...
0
votes
1answer
53 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 ...
8
votes
3answers
504 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 ...
1
vote
0answers
68 views

Characterization of locally conformally flat manifolds: strange application of Frobenius theorem

(Crossposted from math.SE because of the lack of replies) In Hamilton's Ricci Flow (by Chow, Lu, Ni, pp. 29-31, see here) they show that a Riemannian manifold $(M^n,g)$ is locally conformally flat ...
6
votes
1answer
115 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 ...
1
vote
0answers
130 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: $(M_1\times M_2, f^p(g_1 \times g_2))$ ...
9
votes
2answers
3k 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 ...
7
votes
3answers
310 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 ...
40
votes
1answer
1k views

“Gross-Zagier” formulae outside of number theory

The Gross-Zagier formula and various variations of it form the starting point in most of the existing results towards the Birch and Swinnerton-Dyer conjecture. It relates the value at $1$ of the ...