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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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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145 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: $(M_1\times M_2, f^p(g_1 \times g_2))$ ...
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4answers
924 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) ...
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0answers
100 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
324 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 ...
10
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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
146 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
119 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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57 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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48 views

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
75 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
79 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
81 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
76 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
137 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 ...
0
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0answers
79 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
144 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
90 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
41 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 ...
2
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1answer
270 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 ...
2
votes
1answer
141 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 ...
3
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0answers
151 views

Formula for the distance in noncommutative geometry

Probably the most famous formula in noncommutative geometry is the following formula allowing one to compute distance of two points using the operator theoretic data: $$(1) \ \ ...
1
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1answer
94 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 ...
2
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1answer
145 views

Is there a characterization of Riemannian manifolds that split off two factors?

Some Riemannian manifolds are expressed as a product manifold. Recently, I have read two articles about space-times. In both articles, the authors prove that a Riemannian manifold $\bar{M}^n$ is ...
4
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0answers
160 views

Curvature estimates for Kaehler-Einstein and Hermitian, Einstein four-manifolds

Four-dimensional Kaehler-Einstein manifolds and Hermitian, Einstein manifolds with positive scalar curvature have been classified by Tian and LeBrun respectively. I was wondering are there any ...
2
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1answer
149 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, ...
2
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0answers
113 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 ...
0
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1answer
66 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. ...
4
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3answers
310 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 ...
2
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1answer
148 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 ...
2
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0answers
80 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 ...
0
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1answer
71 views

Homotopy bounds in simply connected complete Riemannian manifolds

Let $M$ be a simply connected complete Riemannian manifold, and let $x\in M$. Does there exist a nondecreasing function $R:\mathbb R_+\to\mathbb R_+$ such that, for every $r>0$ and all paths ...
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1answer
145 views

Upper bound of derivative of exponential map

We know that for any simply connected surface $M$,whose Gaussian curvature $K\leq 0$, for any $p\in M$, $exp_p: T_pM\to M$ is diffeomorphism. We know that for any $v\in T_pM$ and $w\in ...
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0answers
140 views

Characterization of the Riemann curvature tensor

Let $(M^n,g)$ be a Riemannian manifold, $a\in M$ be a fixed point. It it well known that there exists a coordinate system near $a$ (e.g. the normal one) such that $$g_{ij}(x)=\delta_{ij}+O(|x|^2).$$ ...
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1answer
152 views

Are there compact Riemannian manifolds whith Q-curvature negative?

Are there known examples of compact Riemannian manifolds with Q-curvature negative?
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1answer
102 views

Obstruction to the existence of global isometries on a constant-curvature Riemannian manifold

Let $M$ be an $m$-dimensional simply connected Riemannian manifold that is not geodesically complete. Suppose $M$ has constant sectional curvature. Because the curvature is constant, locally $M$ ...
2
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1answer
201 views

Do all surfaces (2d riemanian manifolds) admit constant curvature? [closed]

There seems to be a lot of theorems allowing to prove restricted cases of this (eg. uniformization, classification theorem for compact surfaces) . Intuitively, it seems true, but I've never seen a ...
5
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1answer
295 views

Geodesics on manifolds with boundary

Let $(M,g)$ be a Riemannian manifold with non-empty boundary. Is there any notion of injectivity radius on $(M,g)$ in points away from the boundary? By this I mean points lying in $M- \partial M$. ...
5
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2answers
321 views

Alternative proof of Varadhan's formula on Riemann manifolds

Consider Varadhan's famous formula for the kernel of the heat equation on a manifold: $$ \lim_{t \rightarrow 0} t \log h(t,x,y) = - \frac{d(x,y)^2}{4} .$$ I do not have access to his 1967 two ...
1
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1answer
161 views

A geometric property of singular matrices

Let $S\subset M_{n}(\mathbb{R})$ be the singular points of the equation $Det=0$. That is $S$ is the critical points of the determinant function. What matrices belongs to $S$, precisely? Let ...
1
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1answer
197 views

Linearisation of Einstein operator

Let $(M,g)$ be a $(m+1)$-dimensional Riemannian manifold with Levi-Civita connection $\nabla$. The Ricci curvature can be viewed as a differential operator ...
4
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1answer
208 views

The heat kernel as an exponential of an integral

In $\mathbb{R}^n$, if $\gamma$ is a line segment between $x_0 = \gamma (0)$ and $x = \gamma (t)$, one has the following formula: $$\frac {\mathbb{e}^{- \frac{1}{4} \int_0^t <\dot{\gamma}, ...
3
votes
0answers
169 views

“Parallel translate” of a geodesic in the following sense [closed]

Since I'm lazy, I'm shamelessly referring to the following question: Derivative of Exponential Map Given a Riemannian manifold $M$, let $\gamma: (a,b) \to M$ be a geodesic and $E$ a parallel vector ...
2
votes
1answer
75 views

Does the green kernel converge as a series of functions?

Let $(M,g)$ be a compact rimannian manifold. It is well known that we can diagonalyse the Green kernel as a $L^2$ operator acting on functions. Moreover we have the convergence of the following ...
5
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0answers
293 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 ...
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87 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 $\partial U \backslash p$ ...
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0answers
129 views

Closed geodesics avoiding points in hyperbolic surfaces

Let $\Sigma$ be a closed hyperbolic surface. Is it true that for any finite collection of points $x_1,\ldots,x_n\in\Sigma$ there exists a closed geodesic $\gamma$ containing none of them? Remark: It ...
7
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0answers
255 views

Have heat kernels for generalized Laplacians on non-compact manifolds been constructed?

Let $M$ be a non-compact Riemannian manifold which is "nice enough", and $D$ a generalized Laplacian on it. The construction of the heat kernel for the Laplace-Beltrami operator on $M$ seems to be ...
4
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1answer
249 views

Besse p134 Riemann tensor in dimension 4

Does someone have a reference for the proof of 4.72 page 134 of Einstein Manifolds? It is said that $$\check{R}-\vert R\vert^2g/4=S/3 (Ric-S/4) +2\mathring{W}(Ric -S/4) $$ because we are in dimension ...
5
votes
2answers
277 views

Geodesics and Riemannian submersions

Let $X,Y$ be Riemannian manifolds, and $f\colon X\to Y$ be a Riemannian submersion. Let $\gamma$ be a geodesic on $X$ starting at a point $x\in X$ and which is orthogonal to the fiber $f^{-1}(f(x))$. ...