# Questions tagged [curvature]

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

248
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### 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 ...

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2
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### 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 ...

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### Curvature explosion and metric landmark stability

$\newcommand{\prin}{\mathrm{prin}}$Context: Let $S$ be the unit sphere in some finite dimensional vector space $V$. Given a connected compact Lie group real representation $\rho:G\rightarrow O(V)$, ...

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### 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 ...

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### Flat manifolds are local geometric objects

In an article called synthetic geometry in Riemannian manifolds M. Gromov says that
tori (and in general flat manifolds) must be seen as local geometric objects.
He does so after making an example ...

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### 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-...

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votes

1
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### Compact complex non-Kähler manifolds with nef canonical bundle

Are there examples of compact complex manifolds $X$ with $K_X$ nef, but $X$ is not Kähler? Perhaps even non-Moishezon examples?
Here, nef can be defined as follows: For any $\varepsilon>0$ there is ...

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### Expository material on the Gromov-Lawson surgery theorem

I am looking for an expository text on the paper "The classification of simply connected
manifolds of positive scalar curvature" by Gromov and Lawson, in particular on the proof of Theorem A....

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### Asymptotic expansion of the Hessian of the distance function

This question originates from another question. Big thanks to MySheperd whose answer to that question clarified my thoughts.
Let $(M,g)$ be an $n$-dimensional complete Riemannian manifold and $r$ is ...

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### Scalar curvature of homogeneous bounded domains

Is it true that a bounded domain $\Omega \subset \mathbb C^n$ equipped with a homogeneous metric (not necessarily a Bergman metric) has (constant) negative scalar curvature?

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### Hessian of the distance function--comparison with the space form with constant sectional curvature 0

Let $M$ be an $n$-dimensional complete Riemannian manifold and $r$ is the distance function to a fixed point.
The Hessian comparison theorem says that if the sectional curvature of $M$ is bounded (...

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### Definition of sectional curvature of graph and relation to smooth sectional curvature

Let $(M,g)$ be a Riemannian manifold.
Let $\tau$ be a triangulation, i.e. a simplicial complex together with a fixed homeomorphism to $M$.
By forgetting about all cells except $0$-cells and $1$-cells ...

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2
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### Is it known whether a closed simply-connected manifold of non-negative curvature admits positive Ricci?

It is discussed in this question whether a simply-connected closed Riemannian manifold with non-negative Ricci curvature admits positive Ricci curvature, and the answer appears to be "no, there ...

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1
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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 ...

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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 ...

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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,...

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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 ...

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1
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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 ...

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1
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107
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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 ...

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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 ...

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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?

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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 ...

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1
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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 ...

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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 ...

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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
$...

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1
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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 ...

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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 ...

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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 ...

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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 \...

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2
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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 ...

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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
$$\...

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1
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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 ...

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1
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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, ...

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0
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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 \...

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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 ...

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1
answer

143
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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 $...

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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 ...

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1
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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) = \...

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0
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### Derivative of mean curvature and the stability operator

In Thm 1.1 of Huisken's "Evolution of hypersurfaces by their curvature in Riemannian manifolds", it states that for a smooth family of immersions $F(\cdot, t):\Sigma^{n-1}\to M^n$ satisfying ...

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### Mean curvature prescribed rotationally symmetric

Let $\Omega \subset \mathbb{H}^2$ a small disk centred at the origin. I know exist a unique solution $u \in C^{2, \alpha}(\Omega)$ of the system:
$$ H_M = \text{div} \left(\frac{\nabla u}{\sqrt{1+|\...

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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 ...

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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 &...

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### 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 ...

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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 ...

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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}...

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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 ...

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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 ...

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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?...

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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-...

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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 ...