Questions tagged [curvature]

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

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44 votes
2 answers
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Does the curvature determine the metric?

I ask myself, whether the curvature determines the metric. Concretely: Given a compact manifold $M$, are there two metrics $g_1$ and $g_2$, which are not everywhere flat, such that they are not ...
Bernhard Boehmler's user avatar
14 votes
1 answer
1k views

When is a given matrix of two forms a curvature form?

Let's assume we are working over $\mathbb{R}^n$ (but feel free to change to domain to answer the question). I wish to know if the equation $F = dA + A \wedge A$ can be solved for a matrix of 1-forms $...
Vamsi's user avatar
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17 votes
2 answers
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Square of the distance function on a Riemannian manifold

Let $(M^n,g)$ be a smooth Riemannian manifold. Consider the square of the distance function $$dist^2\colon M\times M\to \mathbb{R}$$ given by $(x,y)\mapsto dist^2(x,y)$. It is easy to see that this ...
asv's user avatar
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4 votes
0 answers
486 views

Limit cycles of quadratic systems and closed geodesics(Finitness of $H(2)$)

This question is inspired by this answer to the question Finding a 1-form adapted to a smooth flow. Assume that $V$ is a polynomial vector field of degree $2$ as follows:$$\begin{cases} x'=P(...
Ali Taghavi's user avatar
13 votes
4 answers
6k views

When is a Riemannian metric equivalent to the flat metric on $\mathbb R^n$?

I'm looking for an easily-checked, local condition on an $n$-dimensional Riemannian manifold to determine whether small neighborhoods are isometric to neighborhoods in $\mathbb R^n$. For example, for ...
Theo Johnson-Freyd's user avatar
3 votes
0 answers
161 views

Flat Riemannian metrics adapted to quadratic vector fields with center

Assume that $P(x,y),Q(x,y)\in \mathbb{R}[x,y]$ are two polynomials of degree $2$ with $P(0,0)=Q(0,0)=0.$ Suppose that the vector field $$\begin{cases} x'=P(x,y)\\ y'=Q(x,y) \end{cases}$$ has a center ...
Ali Taghavi's user avatar
20 votes
3 answers
8k views

Curvature of a Lie group

Since a lie group is a manifold with the structure of a continuous group, then each point of the manifold [Edit: provided we fix a metric, for example an invariant or bi-invariant one] has some scalar ...
Matt's user avatar
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12 votes
1 answer
625 views

When is the hull of a space curve composed of developable patches?

Let $C$ be a smooth curve in $\mathbb{R}^3$ that lies entirely on its convex hull, $\cal{H}(C)$. Under what conditions on $C$ is $\cal{H}(C)$ the union of developable surface patches? I believe ...
Joseph O'Rourke's user avatar
3 votes
1 answer
4k views

Relationship between sectional curvature, bisectional curvature and conjugate points

Tian has defined bisectional curvature for unit and perpendicular tangent vectors $X,Y$ as follow $$R(X,Y,X,Y)+R(X,JY,X,JY).$$ If bisectional curvature be constant, is there any relationship between ...
Reza's user avatar
  • 105
1 vote
1 answer
175 views

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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1 vote
3 answers
554 views

Surface analog of clothoid: curvatures covering $\mathbb{R}$

The clothoid $C$, a.k.a. the Euler spiral, is one among many curves with the property that its curvatures cover $\mathbb{R}$ in the sense that, for every $x \in \mathbb{R}$, there is a point $p \in C$ ...
Joseph O'Rourke's user avatar
35 votes
10 answers
8k views

Some questions about scalar curvature

Recall that the scalar curvature of a Riemannian manifold is given by the trace of the Ricci curvature tensor. I will now summarize everything that I know about scalar curvature in three sentences: ...
Paul Siegel's user avatar
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34 votes
7 answers
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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 ...
Gabe K's user avatar
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17 votes
2 answers
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Are there some intrinsic invariants of surfaces other than Gaussian curvature?

The principal curvatures of a surface is denoted by $\kappa_{1}, \kappa_{2}$. Let $P(x,y)$ be a polynomial with real coefficients. Assume that $P(\kappa_{1}, \kappa_{2})$ is an intrinsically ...
Ali Taghavi's user avatar
10 votes
2 answers
899 views

Techniques to solve a non-linear differential equation related to curvature

Many years ago, I considered the following non-linear differential equation: $y=y''\cdot(1+y'^{2})^{-3/2}$ This equation expresses the equality between the value of a given function $y\in C^{2}(R)$ ...
Sylvain JULIEN's user avatar
8 votes
5 answers
3k views

Variation of curvature with respect to immersion?

Let $M$ be a smooth surface and let $f: M \to \mathbb{R}^3$ be a family of immersions given by $$ f(t) = f_0 + tuN_0, $$ where $f_0$ is some initial immersion, $N_0$ is the associated Gauss map, and ...
Omega Tree's user avatar
7 votes
3 answers
460 views

Kernel of a non-integrable connection

The Riemann-Hilbert correspondence states that the kernel of an integrable (zero curvature) connection is a local system. Here, a connexion on a vector bundle $E$ over a manifold $X$ is a morphism of ...
B. Pillet's user avatar
6 votes
1 answer
503 views

minimal surfaces in $S^n$

Thanks to Choi-Schoen theorem, we know that the space of embedded minimal surfaces into $S^3$ of fixed genus is compact. My question are simples: Can we remove the embeddness assumption? Can we ...
Paul's user avatar
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6 votes
1 answer
296 views

Regularization by mean curvature flow

I have a $C^{1,\alpha}$ surface defined as the graph of some function $\varphi : B \to \Bbb{R}_+$ ($B$ is a ball). This surface has positive and bounded mean curvature in the weak sense (since the ...
Beni Bogosel's user avatar
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6 votes
1 answer
323 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
5 votes
1 answer
254 views

Manifolds with boundary admitting no closed embedded minimal hypersurface

The following Theorem is proved in the paper entitled "Compactness of the space of embedded minimal surfaces with free boundary in three-manifolds with nonnegative Ricci curvature and convex ...
Eduardo Longa's user avatar
4 votes
1 answer
1k views

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 (...
Borromean's user avatar
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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
4 votes
2 answers
358 views

Minakshisundaram-Pleijel zeta function identity

Let $\zeta(M,s)$ be the Minakshisundaram-Pleijel zeta function, which encodes the eigenvalues of the Laplace-Beltrami operator. Where can I find a proof or reference of the following identity? If $M$ ...
Nico A's user avatar
  • 447
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
  • 45
3 votes
1 answer
373 views

total mean curvature for singular surface

(a) The total mean curvature of a smooth oriented surface $S$ is defined as $$\iint_S H dA$$ where $H$ is the mean curvature (defined w.r.t. a choice of continuous unit normals on $S$.) Is there a ...
Thomas's user avatar
  • 511
3 votes
1 answer
578 views

Upper bound on the sectional curvature of the orthogonal group

Consider the orthogonal group $O(n)$ as a Riemannian manifold endowed with the usual (bi-invariant) metric $\langle P, Q \rangle_A = \textrm{Tr}\ P^\top Q$ for tangent vectors $P, Q$, with $$T_A O(n) =...
Călin's user avatar
  • 271
3 votes
1 answer
302 views

Manifold with a quasi-positive curvature

As far as I know, in a simply connected compact manifold, still there exists no well-known obstruction for a manifold with a quasi-positive curvature to be a manifold with positive curvature. But ...
Hee Kwon Lee's user avatar
  • 1,070
1 vote
1 answer
171 views

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
  • 6,716