Questions tagged [curvature]

Gaussian curvature, mean curvature, sectional curvature, scalar curvature, curvature tensors (Riemann, Ricci, Weyl)

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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
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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
2 votes
0 answers
114 views

The Ricci curvature is bounded below by scalar curvature

So I have more questions coming from Dr Hamilton's "Three-Manifolds with Positive Ricci Curvature" paper here. I'm working in section 9 on Preserving Positive Ricci Curvature. In theorem 9.4,...
James Chiu's user avatar
5 votes
1 answer
229 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
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Sufficient conditions for the boundary unit-normal vector field of a closed convex set to be Lipschitz continuous

This is followup to the following question: On the Lipschitz continuity of the unit-normal vector field of a polytope Let $C$ be a (nonempty) closed convex subset of $\mathbb R^n$. Note that to every ...
dohmatob's user avatar
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1 vote
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Conditions for Lipschitzness of boundary normal vector, almost everywhere

Let $C$ be a nonempty closed subset of $\mathbb R^n$. It is known that any such set satisfies the following condition (Unique CPP a.e). For almost every $x \in \mathbb R^n$, there exists a unique ...
dohmatob's user avatar
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4 votes
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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
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Knots with everywhere positive curvature

A naive question that my searches have not resolved: Q. Can every knot be realized by a curve in $\mathbb{R}^3$ with strictly positive curvature at every point?
Joseph O'Rourke's user avatar
2 votes
0 answers
98 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
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Heat kernel on hyperbolic space of variable curvature

I am working with the heat kernel on the hyperbolic space explicitly (as you may guess by my previous questions) and I got the desired results when the curvature is $-\kappa=-1$. Now I am trying to do ...
MathqA's user avatar
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Gauss-Bonnet Theorem: Neither Gauss nor Bonnet [closed]

Tristan Needham says (p.174),* "While Gauss and Bonnet certainly paved the road to [the Gauss-Bonnet Theorem], neither one of them was even aware of this extraordinary result, let alone stated ...
Joseph O'Rourke's user avatar
2 votes
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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
8 votes
1 answer
403 views

What should a meaningful notion of curvature satisfy, in the absence of a smooth structure?

There are many generalizations of various curvatures to non-smooth metric spaces (e.g. Ollivier's Ricci curvature). Suppose I have a metric space $(X,d)$ and I want to define a notion of curvature ...
Brendan Mallery's user avatar
5 votes
1 answer
150 views

Is spin cobordism an invariant for surgery of codimension $q\ge3$?

Recall that a surgery of codimension $q$ on an $n$-manifold $X$ is a modification of $X$ of the following type. Let $\Sigma^{n-q}\subset X$ be a smoothly embedded $(n-q)$-sphere with a trivialized ...
wonderich's user avatar
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Are there a geometry behind the singularity formation in solutions to nonlinear ODE's?

Disclaimer: This question was originally posted in math.stackexchange.com and, after 30 days with no answers, I followed the instructions of this topic. If we take two apparently simple first order ...
Diego Santos's user avatar
6 votes
3 answers
330 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
2 votes
2 answers
211 views

Invariant description of the Weitzenböck curvature operator by Bourguignon

I recently came across the paper Les variétés de dimension 4 à signature non nulle dont la courbure est harmonique sont d’Einstein by Jean Pierre Bourguignon. What he shows in §8 is that the ...
Mathy's user avatar
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4 votes
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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
4 votes
2 answers
334 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
3 votes
1 answer
198 views

Ricci curvature of the Weil-Petersson metric?

Let $\omega_{\text{WP}}$ denote the Weil-Petersson metric associated to a family of Calabi-Yau manifolds. That is, let $f : X \to Y$ be a surjective holomorphic map with connected fibres such that, ...
AmorFati's user avatar
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2 votes
0 answers
106 views

Critical points of the area functional restricted to CMC embeddings

For fixed closed smooth manifolds $M^n$ and $N^{n+1}$, two $C^{k,\alpha}$ embeddings $f, f' : M \to N$ are said to be equivalent if there exists $\varphi \in \operatorname{Diff}(M)$ such that $f' = f \...
Eduardo Longa's user avatar
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0 answers
116 views

Is every minimal graph smooth?

The following result was taken from the book of Gilbarg-Trudinger: In particular, if the graph is minimal, then $u$ is smooth. Now comes my question: does the same conclusion hold for graphs over ...
Eduardo Longa's user avatar
0 votes
1 answer
165 views

convergence of the mean curvature under $L^\infty$ norm

Suppose that I have a Jordan curve $J$ parametrized by the function $\phi$. Consider a sequence of parametric functions $\phi_n$ parametrizing a sequence of Jordan curves $J_n$, and denote by $H$ and $...
Jullienne Franz's user avatar
6 votes
2 answers
300 views

Fáry-Milnor theorem for positively curved metrics on $S^3$?

I'm interested in generalizations the following well-known theorem of Fáry and Milnor. Theorem. (Fáry-Milnor) If a simple closed curve $\gamma \subset \mathbb{R}^3$ is knotted, then the total ...
Julian Chaidez's user avatar
1 vote
1 answer
97 views

Closed surfaces of prescribed mean curvature

Let $D\subset\mathbb R^n$ be a smoothly bounded open domain and $0\in D$. For any $x\in\partial D$ there holds \begin{eqnarray*} 2 \,a'(\vert x\vert)\,(x\cdot\nu(x))+(n-1)\,a(\vert x\vert) \, H(x) = \...
guest61's user avatar
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2 votes
0 answers
120 views

PDE for the area-preserving non-parametric curve shortening flow?

In dimension $1$, it is well-known that the motion of a (unparametrized) plane curve $\{(x,f(x,t);\,x \in I \subset \mathbb R)\}$ under the curve shortening flow can be encoded by the PDE $$\partial_t ...
Fei Cao's user avatar
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Intersection of minimal and CMC surfaces

Let $(M,g)$ be a complete and oriented Riemannian 3-manifold without boundary. Given $\Sigma_1$ and $\Sigma_2$ closed surfaces embedded in $M$, where the former is minimal and the latter has CMC $H &...
Eduardo Longa's user avatar
2 votes
0 answers
78 views

Least regularity of boundary to have Lipschitz or bounded mean curvature?

For domains of class $C^{1,1}$, I know that the mean curvature of its boundary belongs to $L^{\infty}(\partial \Omega)$. Is $C^{1,1}$ regularity the least regularity needed for the mean curvature to ...
Jullienne Franz's user avatar
7 votes
1 answer
212 views

Rigidity for convex surfaces in elliptic/hyperbolic space

From Alexandrov's work we know that any metric on the sphere with lower curvature bound $\kappa$ (in the sense of Alexandrov) can be realized as a closed convex surface (i.e. boundary of a compact ...
Pete's user avatar
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5 votes
2 answers
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Taylor expansion of the square of the distance function on a Riemannian manifold [closed]

I have recently read the problem named "Square of the distance function on a Riemannian manifold"(enter link description here) and I am interested in the formula $ d^2(exp_{x_0}(tv),exp_{x_0}...
Luis Yanka Annalisc's user avatar
6 votes
2 answers
405 views

Necessary and sufficient curvature condition for a regular planar curve to be simple and closed

Given a smooth, $2\pi$-periodic function $\kappa(s)$, the associated planar curve $\gamma(s)$ for which $\kappa(s)$ is the (signed) curvature, is uniquely determined up to Euclidean invariance: a ...
Frits Veerman's user avatar
2 votes
1 answer
123 views

Positive scalar curvature on the double of a manifold (assuming mean convexity of the boundary)

This question is related to a previous one. Let $(M^n,g)$ be a compact Riemannian manifold with boundary. Assume it has positive scalar curvature and $\partial M$ is mean convex (positive mean ...
Eduardo Longa's user avatar
4 votes
2 answers
295 views

Positive scalar curvature on the double of a manifold

Let $(M,g)$ be a compact Riemannian manifold with boundary and assume it has positive scalar curvature. Question. Is it true that $DM$, the double of $M$, admits a metric of positive scalar curvature?...
Eduardo Longa's user avatar
1 vote
1 answer
162 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
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1 vote
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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 ...
lady gaga's user avatar
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3 votes
1 answer
298 views

Vanishing Gaussian curvature

I encounter the following claim in my general relativity research: Given a regular surface $x(u,v)$ such that in a neighborhood $V$ os some point $p$ in $S$, the coordinate curves $x(u,v=v_{0})$ and $...
pureorapplied's user avatar
5 votes
2 answers
652 views

Existence of point with zero mean curvature

I'm a physicist studying differential geometry for my GR research, and I come up with the following claim (not sure if it's true or not): For any compact surface $S$ that's not homeomorphic to a ...
geooranalysis's user avatar
2 votes
0 answers
112 views

Canonical class & ring of projective space $\mathbb{P}^n$ in differential geometry

David Mumford remarks in his book Algebraic Geometry I, Complex Projective Varieties on page 109 that the fact that the canonical ring $\oplus_{k=0}^{\infty} \Omega_{k, \mathbb{P}^n}$ of projective ...
user267839's user avatar
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1 vote
0 answers
71 views

Is there an analytic formula (or even a name...) for a plane curve with curvature inversely proportional to x?

I'm interested in plane curves with curvature inversely proportional to distance from the axis: $$\kappa(t) = \left(\frac{x'(t) y''(t) - y'(t)x''(t)}{(x'(t)^2 + y'(t)^2)^{3/2}} \right) = \frac{1}{a x(...
Jacob Schwartz's user avatar
3 votes
1 answer
985 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
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7 votes
2 answers
351 views

Constant Gaussian curvature disks

This question has also been posted on MSE, but maybe here is the right place to post it. Is it true that if $D$ is a Riemannian $2$-disk having constant Gaussian curvature equal to $1$ and whose ...
Eduardo Longa's user avatar
10 votes
1 answer
2k 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
0 votes
0 answers
124 views

Einstein submanifold of Einstein manifold - References

Is there any work in which the conditions under which an Einstein manifold admits an Einstein submanifold are studied? If yes, can you give me the references?
MathDG's user avatar
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34 votes
7 answers
4k views

What is the best way to draw curvature?

This is more of a pedagogical question rather than a strictly mathematical one, but I would like to find good ways to visually depict the notion of curvature. It would be preferable to have pictures ...
Gabe K's user avatar
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5 votes
1 answer
596 views

Sectional curvature of the manifold of symmetric positive definite matrices

I am interested in the sectional curvatures of the manifold of symmetric positive definite $n \times n$ matrices with the affine metric and more precisely in a tight lower bound. It's fairly well ...
Foivos's user avatar
  • 345
4 votes
0 answers
86 views

Curvature universal abelian variety

I am reading N.Mok's paper "Aspects of Kähler Geometry on Arithmetic varieties", I am especially interested in the computation of the curvature for the space $\mathcal{H}_g \times \mathbb{C}^...
user141601's user avatar
1 vote
1 answer
423 views

the curvature wave equation

I was referred here from this question I asked on stackexchange. And now that I'm here, I see that this other question about geometric wave equations is very closely related to mine. But I have a ...
Adam Herbst's user avatar
6 votes
0 answers
118 views

Deriving (Gaussian) curvature bounds from bounds on the metric

I am trying to understand a bound in Christodoulou's 2008 paper on black hole formation. The paper considers a spacelike surface $S$ diffeomorphic to a sphere, with two metrics: the induced metric $\...
Chris's user avatar
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1 vote
0 answers
57 views

Rigidity case of a geometric theorem for $3$-manifolds with boundary

Let $(M^3,g)$ be a compact Riemannian $3$-manifold with boundary. In a paper by L. Ambrozio, he considers the set $\mathcal{F}_M$ of all immersed disks in $M$ whose boundaries are curves in $\partial ...
Eduardo Longa's user avatar
2 votes
2 answers
152 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
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