Questions tagged [curvature]
Gaussian curvature, mean curvature, sectional curvature, scalar curvature, curvature tensors (Riemann, Ricci, Weyl)
284
questions
2
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1
answer
214
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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 ...
5
votes
0
answers
98
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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 ...
2
votes
0
answers
114
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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,...
5
votes
1
answer
229
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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 ...
1
vote
1
answer
171
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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 ...
1
vote
1
answer
175
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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 ...
4
votes
1
answer
239
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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 ...
0
votes
0
answers
101
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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?
2
votes
0
answers
98
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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 ...
1
vote
1
answer
237
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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 ...
15
votes
1
answer
1k
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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 ...
2
votes
0
answers
129
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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
$...
8
votes
1
answer
403
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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 ...
5
votes
1
answer
150
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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 ...
4
votes
1
answer
183
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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 ...
6
votes
3
answers
330
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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 \...
2
votes
2
answers
211
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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 ...
4
votes
0
answers
148
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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
$$\...
4
votes
2
answers
334
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Correct curvature tensor of symmetric space of positive definite matrices with trace metric?
Let $Pos(n)$ be the set of $n \times n$ real positive definite matrices with trace (aka affine-invariant) metric
$$\langle u, v \rangle_p = tr(p^{-1} u p^{-1} v)$$
for all $p \in Pos(n)$ and $u, v \in ...
3
votes
1
answer
198
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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, ...
2
votes
0
answers
106
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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 \...
0
votes
0
answers
116
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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 ...
0
votes
1
answer
165
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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 $...
6
votes
2
answers
300
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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 ...
1
vote
1
answer
97
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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) = \...
2
votes
0
answers
120
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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 ...
2
votes
0
answers
99
views
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 &...
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 ...
7
votes
1
answer
212
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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 ...
5
votes
2
answers
1k
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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}...
6
votes
2
answers
405
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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 ...
2
votes
1
answer
123
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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 ...
4
votes
2
answers
295
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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?...
1
vote
1
answer
162
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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-...
1
vote
0
answers
94
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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 ...
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 $...
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 ...
2
votes
0
answers
112
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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 ...
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(...
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 ...
7
votes
2
answers
351
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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 ...
10
votes
1
answer
2k
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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},...
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?
34
votes
7
answers
4k
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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 ...
5
votes
1
answer
596
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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 ...
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}^...
1
vote
1
answer
423
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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 ...
6
votes
0
answers
118
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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 $\...
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 ...
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(...