Questions tagged [curvature]

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

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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
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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31 votes
2 answers
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Unifying Geometry for Characteristic Classes

When working with characteristic classes (more concretely Chern classes), one finds at least four essentially distinct approaches: Axiomatic Approach. See, for instance, Vector Bundles and K-Theory, ...
Jjm's user avatar
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25 votes
5 answers
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Intuition for mean curvature

I would like to get some intuitive feeling for the mean curvature. The mean curvature of a hypersurface in a Riemannian manifold by definition is the trace of the second fundamental form. Is there ...
nicolas's user avatar
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20 votes
3 answers
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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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19 votes
0 answers
279 views

Modified Willmore energy and surfaces with infinitesimally narrow necks

Disclaimer: This is a copy of a question that I asked on the Mathematics Stack Exchange. It was suggested to me there that the question was worth asking over here. There is an open problem in ...
m3tro's user avatar
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18 votes
1 answer
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Is the minimal volume a topological invariant?

On Wikipedia, it is said that the minimal volume $$\operatorname{MinVol}(M):=\inf\{\operatorname{vol}(M,g) :g\text{ a complete Riemannian metric with }|K_{g}|\leq 1\}$$ is a topological invariant, ...
Cosine's user avatar
  • 559
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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17 votes
3 answers
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Geometric picture of scalar curvature

In first course differential geometry you learn, that Ricci-curvature is something like a mean-value of the curvature endomorphism, because it's a trace, and the scalar curvature is again a mean-value ...
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
17 votes
1 answer
888 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 ...
Tim Campion's user avatar
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15 votes
1 answer
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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
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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14 votes
2 answers
500 views

Do curvature differences obstruct a.e orientation-preserving isometries?

Is there an example of a pair $M,N$ of connected, oriented equidimensional Riemannian manifolds with the following properties: $M$ is everywhere non-flat, $N$ is flat. There exist a map $f:M \to N$ ...
Asaf Shachar's user avatar
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13 votes
4 answers
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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
13 votes
3 answers
2k views

History surrounding Gauss Theorema Egregium and differential geometry

I am teaching a class on elementary differential geometry and I would like to know, for myself and for my students, something more about the history of Gauss Theorema Egregium, that is the Gaussian ...
Giuseppe's user avatar
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0 answers
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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 \...
Tara's user avatar
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12 votes
2 answers
1k views

What is known about Lie groups with (strictly) positive curvature?

If we consider $G$ a compact Lie group, there is a left invariant Riemannian metric whose the sectional curvature is nonnegative (see Milnors' paper). When can we find a left invariant metric that has ...
melomm's user avatar
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12 votes
2 answers
730 views

A Converse to Cartan–Hadamard theorem?

Let $M$ be a complete Riemannian manifold, with the property that $\exp_p\colon T_pM \to M$ is a diffeomorphism for every $p \in M$. Can we say something about it's curvature? Is it true that its ...
Asaf Shachar'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
12 votes
1 answer
419 views

Riemannian vs Non-Riemannian curvature

If you neither know the metric nor the holonomy group, how do you recognize a curvature tensor is Riemannian? I assume a curvature, by definition, satisfies Bianchi identities. I know it is ...
Aureliano Skirzewski's user avatar
12 votes
1 answer
944 views

Differential geometric interpretation of cohomology

I'm not sure whether this is an appropriate question for this forum. I'm afraid that this is not a research level question however: 1. It's about reference request therefore the answer does not ...
truebaran's user avatar
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12 votes
0 answers
247 views

Jacobi fields on non-geodesic curves

The point of Jacobi fields is to study variations of geodesics through geodesics, but the Jacobi equation $D_t^2 J + R(J,\dot\gamma)\dot\gamma=0$ makes sense for any curve $\gamma$, not just for ...
Ethan Dlugie's user avatar
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11 votes
1 answer
940 views

Who first proved that a vanishing Riemann tensor is sufficient for the existence of Euclidean coordinates?

Riemann famously introduced the notion of what we now call a Riemannian manifold and introduced the Riemann curvature tensor $R_{ijk}{}^l$, showing that it is an obstruction the local existence of ...
Igor Khavkine'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
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
10 votes
0 answers
333 views

Which differential forms commute with the curvature form?

Consider a vector bundle, $E \to M$, with connection, $\nabla$, and curvature $2$-form, $F$ on $M$. For $E$-valued differential forms on $M$, $\Omega(M, E)$, we have an exterior covariant derivative, ...
cheyne's user avatar
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9 votes
1 answer
653 views

Constant Gaussian curvature surfaces in 3-space containing lines

How can one construct surfaces in $\mathbb R^3$ of constant negative Gaussian curvature containing a line in $\mathbb R^3$? (this question is inspired by this MSE post).
Mikhail Katz's user avatar
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9 votes
1 answer
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Surfaces in a 3-manifold with the same Gaussian curvature with respect to two ambient conformal metrics

Let $M$ be a 3-smooth manifold and $g_{1}$ and $g_{2}$ two conformal metrics on $M$. Consider an immersed surface S in $M$ and let $K_{1}$ and $K_{2}$ be the Gaussian curvatures of $S$ with respect to ...
Pedro Roitman's user avatar
9 votes
1 answer
950 views

Is there a mathematical explanation for the Aharonov-Casher effect?

Recall that the Aharonov-Bohm effect can be interpreted mathematically as follows. Consider an electromagnetic field A on some smooth manifold M, i.e., A is an element in the first differential ...
Dmitri Pavlov's user avatar
9 votes
1 answer
336 views

Do geodesics avoid regions where the curvature diverges?

Let $(M^2,g)$ be a Riemannian manifold, with manifold boundary $\partial M$. We assume that the metric degenerates at the boundary, in the sense that the (Gauss) curvature diverges like $K \to +\infty$...
Leo Moos's user avatar
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9 votes
0 answers
274 views

Hermitian sectional curvature

Let $N$ be a Riemannian manifold, denote $R$ its purely covariant Riemann curvature tensor with sign convention so that the sectional curvature is $K(X,Y) = R(X,Y,X,Y)$ for an orthonormal pair. ...
seub's user avatar
  • 1,337
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
8 votes
2 answers
441 views

Projective curves of constant curvature

A nodal projective curve in $\mathbb{CP}^2$ inherits a Kähler metric from the Fubini-Study metric, and hence a Riemannian metric. In particular, with respect to this metric, a line has constant ...
Marco Golla's user avatar
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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
8 votes
2 answers
4k views

A question on Ricci curvature and Ricci form.

It seems to me that there are two definition of Ricci flatness; real and complex ones. The real Ricci flatness claims vanishing of Ricci curvature. To define complex Ricci flatness, one needs to ...
Pooya's user avatar
  • 91
8 votes
1 answer
969 views

Injectivity radius on complete manifolds with positive and bounded curvature

I have two question: 1) Are there any examples of complete manifold with strictly positive and bounded section curvature which has zero injectivity radius? 2) Is there a sequence of non-compact ...
Yuchen Bi's user avatar
  • 101
8 votes
1 answer
226 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 ...
Eduardo Longa's user avatar
7 votes
5 answers
4k views

Can anyone give an example of Ricci flat Riemannian or Lorentzian Manifold that is not flat?

Does there exist a Ricci flat Riemannian or Lorentzian manifold which is geodesic complete but not flat? And is there any theorm about Ricci-flat but not flat? I am especially interset in the case ...
346699's user avatar
  • 977
7 votes
2 answers
384 views

Is every metric uniformly close to a metric with negative scalar curvature?

Let $M$ be a smooth manifold with non-empty boundary. Let $g$ be a smooth Riemannian metric on $M$. Is the following true? For every $\epsilon >0$ there exist a Riemannian metric $g_{\epsilon}$ ...
Asaf Shachar's user avatar
  • 6,611
7 votes
3 answers
1k views

Open problems in sub-Riemannian geometry

What are some open problems in sub-Riemannian geometry? I am interested especially in problems concerning connections and curvature, but any contribution is welcomed.
Cristi Stoica'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
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
7 votes
1 answer
193 views

Positively curved manifold with collapsing unit balls

Can we find a complete connected noncompact Riemannian manifold $(M^n,g)$ such that the curvature operator $Rm>0$ and $$ \inf_{p \in M} \text{Vol}_gB(p,1)=0? $$
Totoro's user avatar
  • 2,515
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
  • 115
7 votes
1 answer
1k views

About Sectional Curvature [closed]

In a paper by Yann Ollivier: Let $x$ be a point in $X$, $v$ a small tangent vector at $x$, $y$ the endpoint of $v$, $w_x$ a small tangent vector at $x$, and $w_y$ the parallel transport of $w_x$ from ...
Sepideh Bakhoda's user avatar
7 votes
0 answers
101 views

Is there a convex three-dimensional body with constant width and only finitely-many equilibria? Or: do spheroform gömböcök exist?

Mathematical questions. The mathematical (and 'gravity'-free) formulation of the question in the title is given by the following questions: Q1. Does there exist $(a,b)\in\omega^2\setminus\{(0,0)\}$ ...
Peter Heinig's user avatar
  • 6,001
6 votes
2 answers
570 views

$S^3 \setminus S^1$ doesn't have hyperbolic structure

I need to prove that $M = S^3 \setminus S^1$ doesn't admit any metric of constantly negative sectional curvature s.t. $M$ is complete respect for this metric. I know that it is consequence of famous ...
Mykola Pochekai's user avatar
6 votes
2 answers
741 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. ...
Plank's user avatar
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