Questions tagged [curvature]

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

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2answers
130 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 ...
4
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0answers
141 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 $$\...
4
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1answer
131 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 ...
3
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1answer
112 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, ...
2
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0answers
70 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 \...
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0answers
63 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 ...
0
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1answer
103 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 $...
6
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2answers
184 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 ...
1
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1answer
81 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) = \...
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0answers
56 views

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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0answers
47 views

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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0answers
96 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 ...
2
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0answers
74 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 &...
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0answers
61 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 ...
6
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0answers
128 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 ...
2
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2answers
262 views

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
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2answers
159 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 ...
2
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1answer
70 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 ...
3
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1answer
112 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?...
1
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1answer
110 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-...
1
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0answers
85 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 ...
3
votes
1answer
175 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
2answers
461 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
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0answers
83 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 ...
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0answers
57 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
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1answer
267 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
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2answers
253 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 ...
10
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1answer
677 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},...
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0answers
71 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?
1
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0answers
110 views

Curvature calculation on a holomorphic vector bundle

Let $\mathcal{E} \to M$ be a holomorphic vector bundle over a Kähler manifold. Let $h$ be the Hermitian metric on $\mathcal{E}$. For an endomorphism $A \in \text{End}(\mathcal{E})$, I am trying to ...
29
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7answers
3k 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 ...
3
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0answers
95 views

Geometric interpretation for conformally symmetric manifolds

If $(M, g)$ is a pseudo-Riemannian manifold, it's well known that the condition $\nabla R=0$ is equivalent to the existence of local geodesic symmetries about each point. Another natural algebraic ...
1
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0answers
101 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 ...
4
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0answers
75 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}^...
0
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1answer
108 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 ...
6
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0answers
79 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 $\...
1
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0answers
54 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
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2answers
127 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(...
8
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1answer
125 views

Separating spheres in $3$-manifolds of positive scalar curvature and mean convex boundary

Recently, A. Carlotto and C. Li proved a complete topological classification of those compact, connected and orientable $3$-manifolds with boundary which support Riemannian metrics of positive scalar ...
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0answers
332 views

Algebraic content of Gauss's Theorema Egregium

Let me first recall what Gauss's Theorema Egregium says. Consider a surface isometrically embedded in $\mathbb{R}^3$. In some local coordinates, let the first and second fundamental forms be $$E \...
5
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1answer
163 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 ...
16
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1answer
511 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 ...
1
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1answer
96 views

Wasserstein space with strictly non-positive sectional curvature

Let $(X,d)$ be a (separable, complete) metric space with uniformly strictly non-positive curvature in the sense of Alexandrov, i.e. $(X,d)$ satisfies a $CAT(K)$ inequality for some $K<0$. Does it ...
3
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1answer
298 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 ...
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0answers
69 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 ...
3
votes
1answer
209 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 &...
0
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0answers
124 views

Curvature without any reference to embedding or coordinates

On a 2 sphere, which is topologically just a compact simply connected 2 dim space, one can edow geometric structure by requiring each two geodesics have equidistant intersections. Is this enough to ...
4
votes
1answer
203 views

Positive scalar curvature on the total space of a circle bundle

Let $(\Sigma_\gamma,g)$ be a closed and orientable Riemannian surface of genus $\gamma \geq 1$, $(M^3,\tilde{g})$ be a closed, connected and orientable Riemannian $3$-manifold, and $\pi : M \to \...
4
votes
0answers
72 views

Closed curves with minimal total curvature in the unit circle

Chakerian proved in this paper that a closed curve of length L in the unit ball in $\mathbb{R}^n$ has total curvature at least L. In this later paper Chakerian gave a simpler proof and noted that ...
6
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2answers
651 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. ...

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