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2 votes
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Diameter bounds by mean curvature and area

I'm wondering about a generalization of Simon/Topping/Wu-Zheng's results on bounding diameter by the mean curvature, which roughly says: given a closed $\Sigma^{n-1} \subseteq M^n$, $$\text{diam}(\...
JMK's user avatar
  • 337
1 vote
0 answers
122 views

Bilipschitz constants of exponential map on small ball for Riemannian manifold with curvature bounds

Let $(M,g)$ be a Riemannian manifold with sectional curvature $\mathrm{sect}$ between $-K\le \mathrm{sect} \le K$ for some $K>0$. In [1] it is stated at the beginning of section 4, that if $u,v\in ...
Plamy's user avatar
  • 111
5 votes
0 answers
445 views

Upper bound on the sectional curvature of a Riemannian submersion

Consider the manifold $M := \operatorname{SO}(n) \times \mathbb{S}^{n-1}$, endowed with the product metric given by the bi-invariant metric of $\operatorname{SO}(n)$ and the round metric of $\mathbb{S}...
mathusername's user avatar
14 votes
1 answer
1k views

Progress on Gromov's Conjecture of the bound of total Betti numbers

This question is a reference request. Let $(M,g)$ be a Riemannian manifold of dimension $n$, and $b_i(M) = \dim H_i(M,\mathbb{R})$. Gromov proved it that there are constants $C(n)$ such that, if the ...
fffmatch's user avatar
  • 175
0 votes
1 answer
74 views

Handling degenerate planes in pseudo-Riemannian geometry: impact on sectional curvature and comparison theorems

I've been studying Riemannian and pseudo-Riemannian manifolds and came across an intriguing point regarding the definition of sectional curvature in both geometries. In pseudo-Riemannian geometry, for ...
lming2's user avatar
  • 45
2 votes
0 answers
123 views

Asymptotics on the number of diffeomorphism classes in the Cheeger finiteness theorem

A result of Cheeger says that, given any even dimension and any $\delta > 0$, there are only a finite number of diffeomorphism types of compact simply-connected manifolds of that dimension which ...
macbeth's user avatar
  • 3,212
4 votes
0 answers
167 views

Parallel transport of global sections and Riemannian curvature

A, perhaps, naive question from an algebraist/combinatorialist teaching differential geometry. Originally asked on math.SE but didn't receive a single comment in 3 days. Consider a (real) smooth ...
Igor Makhlin's user avatar
  • 3,513
5 votes
1 answer
343 views

Clarifying a result of Klingenberg

I am looking for some clarification on a result of Klingenberg. For context, I have a complete Riemannian manifold for which I would like to compute a lower bound on the injectivity radius at each ...
E G's user avatar
  • 163
3 votes
0 answers
102 views

Geometric interpretation for a connection whose corresponding distribution generates the whole Lie algebras of vector fields

Let we have a connection on a manifold $M$ so it is considered as a distribution on the tangent bundle $TM$ of $M$. The integrability of this distrbution is equivalent to flatness of the connection. ...
Ali Taghavi's user avatar
3 votes
0 answers
165 views

A higher-dimensional "line of curvature"?

Let $M$ be a submanifold of a Riemannian manifold $Q$, and let $\Gamma$ be a $d$-dimensional submanifold of $M$, i.e., $\Gamma^{d} \subset M \subset Q$. Suppose that, for all (unit) normal vectors of $...
Matteo Raffaelli's user avatar
0 votes
1 answer
100 views

Curvature tensor of interpolation of two metrics

Let $\hat{g}$ and $\bar{g}$ be two smooth Riemannian metrics defined, say, on $\mathbb{R}^n.$ Consider a smooth function $\xi$ that acts as an interpolation function between the two metrics above on ...
Lucas L.'s user avatar
1 vote
0 answers
210 views

Ricci-flat metrics on complex tori of dimension $n \geq 3$

Let $\mathbb{T}^n = \mathbb{C}^n /\Lambda$ be a complex torus of (complex) dimension $n$. If $n=2$, it is a theorem of Berger that the Ricci-flat metrics on $\mathbb{T}^2$ are flat. This follows from ...
AmorFati's user avatar
  • 1,379
0 votes
1 answer
117 views

Curvature of an affine system

I find an interesting paper that mentioned the Definition of curvature of an affine optimal control system. It reminded me that many textbooks on Riemannian geometry only tell us about metrics, ...
lumw's user avatar
  • 111
18 votes
1 answer
1k views

Is the minimal volume a topological invariant?

On Wikipedia, it is said that the minimal volume $$\operatorname{MinVol}(M):=\inf\{\operatorname{vol}(M,g) :g\text{ a complete Riemannian metric with }|K_{g}|\leq 1\}$$ is a topological invariant, ...
Cosine's user avatar
  • 609
5 votes
0 answers
244 views

Möbius strip zero curvature [closed]

Is there a Möbius strip, seen as an embedded surface in $\mathbb{R}^3$, with zero curvature? I know one can see the Möbius strip as the quotient of the square with reverse identification of two sides ...
ibroketheinternet's user avatar
2 votes
0 answers
126 views

Conformal changes of metric and normal coordinates

Suppose that $(M,g)$ is a smooth Riemannian manifold of dimension $n\geq 2$. Let $p\in M^{\textrm{int}}$. Does there exist a small $\delta>0$ and a smooth function $c>0$ such that for the ...
Ali's user avatar
  • 4,145
5 votes
2 answers
340 views

Example of a curvature with no associated metric

Is there a concrete example of a $4$ tensor $R_{ijkl}$ with the same symmetries as the Riemannian curvature tensor, i.e. \begin{gather*} R_{ijkl} = - R_{ijlk},\quad R_{ijkl} = R_{jikl},\quad R_{ijkl} =...
Matthew Lou's user avatar
4 votes
1 answer
439 views

Etymology “Kulkarni–Nomizu product”

$\newcommand\KN{\mathbin{\bigcirc\mspace{-20mu}\wedge\mspace{3mu}}}$In the context of (pseudo)-Riemmian geometry, the Kulkarni–Nomizu product is defined to be an operation $\KN$, which takes two ...
G. Blaickner's user avatar
  • 1,429
9 votes
1 answer
344 views

Do geodesics avoid regions where the curvature diverges?

Let $(M^2,g)$ be a Riemannian manifold, with manifold boundary $\partial M$. We assume that the metric degenerates at the boundary, in the sense that the (Gauss) curvature diverges like $K \to +\infty$...
Leo Moos's user avatar
  • 5,038
1 vote
2 answers
284 views

Ricci scalar of submanifold of $\mathbf R^n$

Let $M$ be an $n-m$ dimensional sub-manifold of $\mathbf R^n$ defined by the following set of equations: \begin{equation} f_1(\vec x)=0, \\ \vdots \\ f_m(\vec x)=0, \end{equation} where $\vec x$ are ...
dennis's user avatar
  • 521
1 vote
2 answers
148 views

Construct a hypersurface with fixed principal curvatures at a point

I'm reading Eschenburg's paper Local convexity and nonnegative curvature — Gromov's proof of the sphere theorem recently. And I meet a little question: Given a point $p\in M$, $N\in T_pM$, we want to ...
eulershi's user avatar
  • 241
0 votes
0 answers
252 views

Computation of scalar curvature from a Riemannian metric

I want to compute the scalar curvature for points on an empirical manifold (sampled data). I have already an algorithm that learns the Riemannian metric and computes geodesics, so from the metric I ...
can't stop me now's user avatar
6 votes
1 answer
463 views

Holonomy bounded in terms of area and the curvature

I suppose the following result follows from Ambrose-Singer theorem, but I cannot find a reference, and the arguments I found in the literature are usually weaker. The idea is that holonomy over a null-...
Misha Verbitsky's user avatar
2 votes
1 answer
224 views

The differentiability of the distance function on asymptotically flat manifolds

Let $M = \mathbb{R}^3 \setminus \overline{B_1}$ where $\overline{B_1}$ is the closed unit ball. Let $g$ be an asymptotically flat metric of the form $g_{ij} = \delta_{ij}+h_{ij}$ in standard ...
Laithy's user avatar
  • 969
5 votes
0 answers
101 views

How is this product of tensors defined?

I am reading the paper “ The first eigenvalue of a small geodesic ball in a riemannian manifold”, by Karp and Pinsky, from where I took the following: Here, $\Delta_{-2}$ denotes the usual Laplacian ...
Eduardo Longa's user avatar
5 votes
1 answer
245 views

Possible sign of scalar curvature for Einstein warped product manifold with Ricci-flat

Let $(M, g_M)$ where $M= B \times_f F$ and $g_M=g_B + f^2g_F$, an Einstein warped product manifold (i.e., $Ric_M= \lambda g_M$), with Ricci flat fiber-manifold $F$, i.e., $Ric_F=0$. Then $M$ can admit ...
MathDG's user avatar
  • 272
4 votes
1 answer
245 views

Tzitzeica surface

A Tzitzeica surface has the property that the ratio of the surface’s Gaussian curvature and the fourth power of the distance from the origin to the tangent plane at any arbitrary point of the surface ...
MathDG's user avatar
  • 272
2 votes
0 answers
101 views

Parallelism defect

I have a question that I don't know how to answer. If I have a parallelism defect it is due to the presence of a curvature and therefore we can bring it back to a Riemann tensor. The thing that is not ...
MathDG's user avatar
  • 272
2 votes
0 answers
149 views

Comparison of sum of vectors and exponential map on a Riemannian manifold

Suppose $M$ is a simply-connected complete Riemannian manifold with bounded sectional curvature $\delta \leq K \leq \Delta < 0$. Let $p_0\in M$. Define the sequence of points $p_1, \ldots, p_n$ by $...
M.R.Karimi's user avatar
6 votes
3 answers
368 views

Curvature function as a random variable with uniform distribution

Let $(S,g)$ be a Riemannian surface. Then the curvature function $\kappa: S\to \mathbb{R}$ can be counted as a random variable. So it produces a probability density function $f_g:\mathbb{R}\to \...
Ali Taghavi's user avatar
4 votes
0 answers
148 views

The range of $\int_M \kappa_g ds$ where $g$ varies in all possible real analytic metrics on $M$

Let $M$ be a real analytic open surface(A non compact 2 dimensional manifold without boundary). For every number $\lambda\in \mathbb{R}$, is there a real analytic Riemannian metric on $M$ with $$\...
Ali Taghavi's user avatar
5 votes
2 answers
379 views

Correct curvature tensor of symmetric space of positive definite matrices with trace metric?

Let $Pos(n)$ be the set of $n \times n$ real positive definite matrices with trace (aka affine-invariant) metric $$\langle u, v \rangle_p = tr(p^{-1} u p^{-1} v)$$ for all $p \in Pos(n)$ and $u, v \in ...
ccriscitiello's user avatar
1 vote
1 answer
178 views

Asymptotics of constant mean curvature surfaces

Let $\Sigma^n \subset \mathbf{R}^{n+1}$ be a complete, properly embedded hypersurface with constant, non-zero mean curvature $H \neq 0.$ In the case where the dimension is $n = 2$, $\Sigma$ is non-...
Leo Moos's user avatar
  • 5,038
1 vote
0 answers
97 views

Computing/estimating geodesics in practice

Let's say I have a Riemannian manifold with known metric $g$. I want to compute the geodesics of this manifold, specifically with respect to the Levi-Civita connection. In practice, (i.e. with a ...
900edges's user avatar
  • 153
4 votes
1 answer
1k views

Relation between mean curvature and conformal metric

We'll consider $(N, g)$ a Riamannian Manifold and $\overline{g} = e^{2f}g$ a conformal metric. Let M be a hypersurface in N, $\overline{H}_M$ and $H_M$ the mean curvature of M with respect to the ...
K2-liz's user avatar
  • 55
10 votes
1 answer
3k views

Taylor expansion of the metric tensor in the normal coordinates

I am looking for a reference with a Taylor expansion of the metric tensor in the normal coordinates. The coefficients should be written in terms of $\mathrm{Rm}, \nabla\mathrm{Rm}, \nabla^2\mathrm{Rm},...
Anton Petrunin's user avatar
2 votes
2 answers
163 views

stability of two-sided sectional curvature bounds in Lorentzian geometry

Suppose that $(M,g)$ is a Lorentzian manifold of signature $(-,+,\ldots,+)$. Given a two plane $\Pi=\textrm{Span}\{X,Y\}$ with $X,Y \in T_pM$, we say that $\Pi$ is non-degenerate if $$ g(X,X)g(Y,Y)-g(...
Ali's user avatar
  • 4,145
18 votes
1 answer
1k views

Geometric interpretation of the Weyl tensor?

The Riemann curvature tensor ${R^a}_{bcd}$ has a direct geometric interpretation in terms of parallel transport around infinitesimal loops. Question: Is there a similarly direct geometric ...
Tim Campion's user avatar
  • 63.9k
3 votes
1 answer
370 views

Reference request: extendability of Lipschitz maps as a synthetic notion of curvature bounds

In the lecture Notions of Scalar Curvature - IAS around 8:00, Gromov states the following result, which he claims he does "slightly uncarefully": Suppose $(X,g_X)$ and $(Y,g_Y)$ are ...
Lawrence Mouillé's user avatar
0 votes
0 answers
126 views

mean curvature for codimension $>1$?

The mean curvature of a hypersurface in a Riemannian manifold is defined to be the trace of the second fundamental form. I was curious, does the notion of mean curvature generalise to higher ...
Johnny T.'s user avatar
  • 3,625
3 votes
1 answer
551 views

Product formula for Laplace de-Rham operator

Let $M$ be a Riemannian manifold with Laplace de-Rham operator $\Delta = (d + \delta)^2$. If $g$ is a smooth $k$-form, and $f$ is a smooth function, is there a simple formula for $\Delta(fg)$ when $k &...
Nathanael Schilling's user avatar
6 votes
2 answers
753 views

Curvature of nonsymmetric metric tensors?

Consider a smooth manifold $M$ of arbitrary dimension. We have notions of psuedo-Riemannian or Riemannian metrics on a manifold, and they differ in the slightest way of being positive-definite or not. ...
Plank's user avatar
  • 327
12 votes
0 answers
262 views

Jacobi fields on non-geodesic curves

The point of Jacobi fields is to study variations of geodesics through geodesics, but the Jacobi equation $D_t^2 J + R(J,\dot\gamma)\dot\gamma=0$ makes sense for any curve $\gamma$, not just for ...
Ethan Dlugie's user avatar
  • 1,277
7 votes
1 answer
197 views

Positively curved manifold with collapsing unit balls

Can we find a complete connected noncompact Riemannian manifold $(M^n,g)$ such that the curvature operator $Rm>0$ and $$ \inf_{p \in M} \text{Vol}_gB(p,1)=0? $$
Totoro's user avatar
  • 2,535
5 votes
1 answer
156 views

Positively curved metric with uniformly positive scalar curvature

Can we find a complete noncompact Riemannian manifold $(M^n,g)$ with bounded geometry satisfying the following conditions? the curvature operator $Rm>0$; the scalar curvature $R \ge 1$. Notice ...
Totoro's user avatar
  • 2,535
4 votes
0 answers
880 views

Scalar curvature in terms of second fundamental form, reference request

I would like to cite a reference for the following formula for scalar curvature: If $\Sigma$ is a hypersurface in Euclidean space, then $R=H^2-\lvert A\rvert^2$, where $R$ is the scalar curvature ...
Yasha Berchenko-Kogan's user avatar
0 votes
0 answers
55 views

Gauss curvature of a fibre as a submanifold in a Riemannian warped product

Consider the Riemannian warped product $M^{n+1}=I\times\mathbb{S}^n$ with metric \begin{align} g=dt\otimes dt+f(t)^2g_{\mathbb{S}^n} \end{align} where $I\subseteq\mathbb{R}$ is some open interval and ...
Anonymous amateur's user avatar
6 votes
1 answer
378 views

Unbounded sectional curvature implies infinite diameter?

Let $(M,g)$ be a Riemannian manifold such that for each $C>0$ there is $p\in M$ and $X,Y\in T_pM$ unitary such that $K(X,Y) > C.$ Does this imply that the diameter of $(M,g)$ is infinite? I ...
L.F. Cavenaghi's user avatar
7 votes
0 answers
115 views

The space of positive scalar curvature metrics on $S^4$

Let $\mathcal{R}_{+\mathrm{sc}}(S^n)$ denote the space of complete Riemannian metrics of positive scalar curvature on the sphere $S^n$. It's known that $\mathcal{R}_{+\mathrm{sc}}(S^2)$ is ...
Tyrone's user avatar
  • 5,596
7 votes
0 answers
1k views

Conventions for Riemann curvature tensor

I am aware of two conventions for the Riemann curvature tensor, namely the expression $$\langle\nabla_X\nabla_YZ-\nabla_Y\nabla_XZ-\nabla_{[X,Y]}Z,W\rangle$$ is either declared to be $R(X,Y,Z,W)$ or $...
John Pardon's user avatar
  • 18.7k