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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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 ...
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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 ...
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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 ...
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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 ...
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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} ...
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183 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 ...
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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 ...
4
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1answer
246 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 ...
6
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2answers
206 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
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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?
2
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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 ...
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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 ...
2
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2answers
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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 ...
5
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0answers
184 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 ...
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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) $ ...
11
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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 ...
2
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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 ...
0
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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
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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 ...
0
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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
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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 ...
5
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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 ...
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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 ...
5
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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 ...
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3answers
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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 ...
4
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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 ...
5
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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 ...
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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 ...
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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
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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 ...
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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
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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 ...
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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: $(M_1\times M_2, f^p(g_1 \times g_2))$ ...
15
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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) ...
3
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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 ...
7
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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 ...
9
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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 ...
3
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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 ...
2
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1answer
111 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 ...
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0answers
55 views

Derivative of Log_p map

If $\mathscr{M}$ is a $(d-dimensional)$ Riemann Manifold and $p$ is a point therein. What is the derivative of $Log_p$ function?
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Optimal bound in $L^2$ product on compact Kahler manifold

Let $X$ be a compact Kahler manifold of dimension $n$, equipped with a Kahler metric of volume $1$. There exists a constant $C \geq 1$ such that for any smooth functions $f,g$ on $X$ we have $$ \int_X ...
2
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0answers
74 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 ...
4
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0answers
78 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 ...
2
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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 $\varphi : U \rightarrow D$ ...
3
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1answer
74 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 ...
3
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1answer
130 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 ...
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0answers
77 views

Functional involving Ricci curvature: convex and coercive?

Let $(M,g)$ be a Riemannian manifold with Levi-Civita connection $\nabla$, volume form $\mu_g$, and Ricci curvature $\text{Rc}_g$. Question: Given a fixed vector field $V\in\Gamma(TM)$, under what ...
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1answer
139 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 $G\subset Isometries(M,g)$ ...
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0answers
88 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) ...
2
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0answers
37 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 ...