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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Length spectrum for Riemannian metrics in the projective plane

Are there (known) examples of non-isometric Riemannian metrics on the projective plane that have the same length spectrum? This question is related to MO questions Length spectrum and Zoll surfaces ...
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208 views

Length spectrum and Zoll surfaces of revolution

The earlier MO question, "Length spectrum of spheres," asked if the length spectrum of closed geodesics determines the metric on $S^2$, and the answer was a clear No due to Zoll surfaces, all of whose ...
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254 views

Is it overkill to invoke Kirszbraun theorem to prove the following fact ?

Given a small enough convex triangle $(abc)$ in a (smooth or Alexandrov) surface $(X,d)$ of curvature greater than $-1$, let $(\overline{abc})$ be its comparison triangle in $\mathbb{H}^2$. Then there ...
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1answer
654 views

Integration By Parts on Non-compact Manifolds

This is undoubtedly a very easy question, but perhaps there are some subtleties. Under what circumstances can we integrate by parts over a non-compact Riemannian manifold? I am aware that having ...
7
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471 views

Full isometry groups of Stiefel and Grassmann manifolds

Hi, I'm looking for a reference for the full isometry groups of the (i) complex Stiefel manifolds $U(m)/U(m-l)$, either for the Euclidean metric (i.e. identifying it with orthonormal $m \times ...
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2answers
335 views

Conformal structure does not see conical singularities

the conformal structure does not see the conical singularities of a polyhedral surface. This is a quote from the Preface of Quantum Triangulations (eds.: Carfora, Marzuoli). The sentiment is ...
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329 views

Best metrics on exotic R^4

What is known about the existence of complete metrics with good properties (e.g., Einstein, constant scalar curvature, etc...) on exotic ${\bf R}^4$s? Note, that some exotic ${\bf R}^4$s have ...
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230 views

What's the relationship between the riemannian metric and Jacobi field?

I encounter to the question in reading the following Excise: Let $(M,g)$ be a $m$-dimensional Riemannian manifold, and $(r,\theta^1,\theta^2,\ldots,\theta^{m-1})$ be the (geodesic) polar ...
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3answers
676 views

Is there a coordinate-free proof of the hamiltonian character of the geodesic flow?

I do not know if this question is appropriate for this site, but I posted here without having answers, so now I make this attempt. Let be $(M,g)$ a pseudoriemannian manifold. Let us identify the ...
4
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1answer
806 views

Fubini Study Metric and Einstein constant

Hi all, it is well known that the complex projective space with the fubini study metric is Einstein, but what is the explicit value, i.e. for which $\mu$ does $Ric=\mu g$ hold? Moreover, I would ...
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246 views

einstein metrics on the tangent bundle

hi, i have the following question. let $M$ be a compact, real analytic, riemannian manifold with real analytic metric $g$. does the tangent bundle admit always a einstein metric ? marco
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268 views

Cheeger's Finiteness Theorem and Lipschitz Constant

Cheeger's Finiteness Theorem states that For each positive numbers $D,v,n$, the number of diffeomorphism classes of Riemannian manifolds $M$ with $Diameter(M)\le D$, $Vol(M)\ge v$, and $|K(M)|\le 1$ ...
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183 views

Seek “typical examples” for the structure of spaces with two-sided Ricci bounds

By a 1990 paper of Michael Anderson, the following is true: Theorem. Let the metric space $(X,d,p)$ be a pointed Gromov-Hausdorff limit of a sequence of complete pointed Riemannian manifolds ...
3
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4answers
1k views

space of geodesics

hallo, i have the following problem: Let $(M,g)$ be a compact Riemannian manifold with metric $g$ and $\nabla$ be the Levi-Civita Connection. Denote by $G(M) =${$\gamma: \mathbb{R} \rightarrow M | ...
4
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1answer
471 views

How the Jacobi metrics may be useful in mechanics with or without constraints?

A mechanical system $(Q,K,V)$ is specified by the configuration space $Q,$ the potential energy $V\in C^\infty(Q),$ and the kinetic energy $K=K_g$ given by a Riemannian metric $g$ on $Q.$ If ...
5
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1answer
463 views

Partitions of Unity

Fix a metric $g$ on a smooth, closed manifold $\mathcal{M}$. Take a finite subcover of the manifold from its atlas. Is it true that any smooth partition of unity subordinate to this cover has ...
4
votes
2answers
436 views

Which vector bundle are the Christoffel symbols sections of?

The collection of Christoffel symbols $\Gamma_{ij}^k$ of a connection (or of a metric) on a smooth manifold $M$ is not the collection of components of a tensor field in some local chart, i.e. they ...
4
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1answer
211 views

Symmetries vs. Bound in codimension of Nash isometric embedding

Let $(M^m,g)$ be a compact smooth Riemannian manifold of dimension $m$. From the celebrated Nash Embedding Theorem, we know there exists a (smooth) isometric embedding $M\hookrightarrow\mathbb R^n$ on ...
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732 views

Metric Connections on a Lie Group

A Lie group has three standard Cartan connections; the (-)-connection, the (0)-connection, and the (+)-connection. The (0)-connection is Levi-Civita with the associated metric the bi-invariant metric. ...
7
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2answers
860 views

Existence, uniqueness, and regularity for linear parabolic PDE on a complete Riemannian manifold

Let $M$ be a smooth manifold with a complete Riemannian metric $g$ and $E$ a smooth vector bundle over $M$ with an inner product and compatible connection $\nabla$. Let $K: E \rightarrow E$ be a ...
6
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0answers
265 views

Compactness of solutions to parabolic equations (parabolic regularity)

I am working on a Ricci flow $(\mathcal{M} , g(t))$, for which the conjugate heat operator is $\Box ^* := - \partial _t - \Delta + R$, where $R$ is the scalar curvature. For each $s>0$, I have a ...
3
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1answer
326 views

eigenspinors of Dirac operator

$M$ compact manifold. Let $\lambda$ be an eigenvalue for the Dirac operator of multiplicity greater than 2. I'm interested in showing the existence of two linearly independant eigenspinors $u$ and $v$ ...
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4answers
736 views

Riemannian metric on a flag variety

$\def\C{\mathbb{C}}\def\CP{\mathbb{CP}}$Every complex projective space $\CP^n$ has a natural Riemannian metric, the Fubini–Study metric, which is defined via the quotient definition of $\CP^n = ...
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167 views

Methods for generating metrics and minimizing variational dynamics of particles (masses or charges) on n-dimensional smooth manifolds

I am attempting to investigate transformations between two distinct sets of vertices on n-dimensional manifolds with a minimal change in the fundamental shape of the vertices. I will give some ...
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3answers
280 views

Large geodesically convex subsets of tori

Let $X=\mathbb T^d=\mathbb R^d/\mathbb Z^d$ and let $E$ be a proper open subset of $X$. We say $E$ is geodesically convex if for any $x,y\in E$ the shortest geodesic connecting $x$ and $y$ lies in ...
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Behavior of sectional curvature under metric deformations

Metric deformation: Let $(M,g_0)$ be a Riemannian manifold and consider a (sufficiently smooth) deformation of $g_0$, $$g_t=g_0+th+O(t^2), \quad 0< t<\varepsilon $$ where $h$ is some symmetric ...
2
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1answer
306 views

Conformally-flat

Assume given a smooth manifold $(\mathbb{R}^n, g)$, where the metric is a scaled identity $g = e^{2f}I$. Is there a way to know if this is always a non-positive (sectional) curvature manifold? Note ...
5
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1answer
918 views

Good Surface,Bad Surface-Surface classification

Maybe this question be very simple, but I don't know why it is hard for me. Thanks for any guide and help. We say a surface $S$ (2-dimensional metric(compact) Riemannian surface) is good (denote by ...
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2answers
341 views

intersection of geodesiques

Let $(M,g)$ be a closed riemannian surface . let $\alpha$ be a simple closed geodesique . does there is exist a simple closed geodesic $\beta$ that intersect alpha at only 1 point p such that ...
4
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5answers
501 views

Exponential and Logarithm Mapping on Stiefel Manifold

The Stiefel Manifold is defined as $$ \mathrm{St}(p,n):= \{ X\in \mathbb{R}^{n\times p} :\ X^T X = I_p \}. $$ Recall that the tangent space at a point $X\in \mathrm{St}(p,n)$ is given by $$ ...
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0answers
107 views

rigidity of eigenvalues of circular ensemble

Given a circular unitary ensemble, with the following joint density: $p(\theta_1,\ldots, \theta_n) = Z_n \prod_{j < k} |e^{i \theta_j} - e^{i \theta_k}|^2$, is the following statement true? With ...
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3answers
497 views

volume of compact simple Lie groups under the natural Euclidean embedding

I am looking for a quick reference for the volume formula for all the compact simple Lie groups embedded as matrix groups in the natural way. The one I care most for are the real orthogonal groups. I ...
4
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1answer
617 views

Length spaces with continuous length functional: is this set Gromov-Hausdorff closed?

As far as I can tell, a major motivation for the study of length spaces is that they arise as Gromov-Hausdorff limits of Riemannian manifolds. Specifically, A complete connected Riemannian manifold ...
3
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1answer
283 views

extended forms from foliations [closed]

hi, i have the following question: Let $M$ be a n-dimensional manifold (or riemannian or everything thats nice ...) and let $\mathcal{F}$ be a foliation of $M$ by surfaces. Assume, furthermore, that ...
3
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2answers
197 views

non commutative elements in the fundamental group of a closed hyperbolic surface

Let $(M,hyp)$ be a closed hyperbolic surface. fix a point $m$ in $M$ and denote by $G=\pi_1(M,m) $. now let $\alpha$ and $\beta$ in $G$ such that $\alpha$ and $\beta$ does not commute . my first ...
9
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2answers
659 views

Good reference for globally formulated calculus of variations on Riemannian manifolds?

I have been working on a formulation of the calculus of variations on Riemannian manifolds, formulated globally using [pullback] vector bundles and tensor bundles and their induced covariant ...
9
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0answers
385 views

Killing spinors and symmetric tensor fields.

Hi all, I have a question of the following form: Let $(M,g)$ be a Riemannian spin manifold which admits a Killing spinor $\sigma$ and let $h:T M \to T M$ be a symmetric, trace-free and ...
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1answer
102 views

Minimal representative of the elements of the fundamental group of a negatively curved manifold

Let (M,g) be a negatively curved manifold , let p be any point of M and denote by G=π1(M,p) . the minimal representative (by minimal i mean the smallest length representative ) of every α in G is a ...
10
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1answer
953 views

Theorem of Bryant in higher dimensions

hallo, i have the following question. i read about Bryant's theorem which sais that: any real-analytic 3-dimensional Riemannian manifold $(Y,g)$ with real-analytic metric $g$ can be isometrically ...
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2answers
461 views

Riemann surfaces with bounded curvature

Say there are metrics $g_n$ on a compact Riemann surface $\Sigma$ with bounded curvature and bounded area, or even with the same area element . What can we say about the 'limit' of $(\Sigma, g_n)$? ...
0
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2answers
294 views

norm of n-th covariant derivative of smooth function

The question is how define the norm of n-th covariant derivative of smooth function f on a manifold M. The manifold is two dimensional so maybe I can do it in the following way: thing about n-th ...
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4answers
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What are “good” examples of spin manifolds?

I'm trying to get a grasp on what it means for a manifold to be spin. My question is, roughly: What are some "good" (in the sense of illustrating the concept) examples of manifolds which are spin ...
2
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1answer
251 views

The Tubular Neighborhood of a Closed Geodesic

Suppose $M_{g}$ is the mapping torus $\Sigma_{g} \times [0, 1]/ (x, 0) \equiv (\tau x, 1)$, where $\Sigma_{g}$ is the hyperbolic space with genus $g,$ and $\tau : \Sigma_{g} \to \Sigma_{g}$ is an ...
2
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1answer
572 views

Isometry groups of Riemannian submersions with totally geodesic fibers

Suppose $F\to M\stackrel{\pi}{\to} B$ is a Riemannian submersion with totally geodesic fibers, all manifolds compact. In general, unless $M=B\times F$ is a Riemannian product, the isometry groups of ...
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1answer
236 views

Dimension of certain subgroup of isometry group of positively curved manifold

Let $M$ be a closed $n$-dimensional Riemannian manifold with positive sectional curvature. Let $G$ be a close subgroup of isometry group ${\rm Iso}(M)$. Suppose the action of $G$ on $M$ is not ...
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2answers
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Area of distance sphere in manifold with Ricci $\ge 0$.

Let $M$ be a open complete manifold with Ricci curvature $\ge 0$. By a theorem of Calabi and Yau, the volume growth of $M$ is at least of linear. I am wondering whether the following statement is ...
10
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616 views

Why the quarter in the $\frac{1}{4}$-pinched sphere theorem?

Is there any hope of a high-level explanation of why the fraction $\frac{1}{4}$ plays such a prominent role as a sectional curvature bound in Riemannian geometry? My (dim) understanding is that the ...
2
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1answer
236 views

Minimum set of subharmonic function in $\mathbb R^n$

Let $f :\mathbb R^n\to \mathbb [0, \infty)$ be a (continuous, $C^2$, or smooth) subharmonic function with minimum value $0$. Then we know the sublevel set $f^{-1}((-\infty, c])$ is mean convex for $c ...
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2answers
450 views

Example for Busemann function is not an exhaustion when Ricci $\ge 0$

For an open complete Riemannian manifold $M$ with non-negative sectional curvature, the Busemann function defined below is a convex exhaustion function (by Cheeger-Gromoll's proof of soul theorem) ...
18
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1answer
604 views

Is the following a sufficient condition for asphericity?

I recently came across the following question while working on some problems on manifolds with lower Ricci curvature bounds. Given $n$ does there exist a large $R>0$ with the following property: ...