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9
votes
3answers
923 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 ...
9
votes
1answer
386 views

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 ...
9
votes
1answer
546 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 ...
6
votes
3answers
993 views

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 ...
6
votes
0answers
210 views

Negative curvature in the middle of $R^{3}$

What's a simple example of metric on $R^{3}$ which has negative scalar curvature inside of a (limited) set N, and is equal to the standard (Euclidean) metric outside? Basically, I am asking for a ...
5
votes
5answers
834 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 ...
5
votes
1answer
629 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 ...
3
votes
2answers
806 views

Interpretation of Curvature and Torsion

Dear all, When dealing with General Relativity one uses the Levi-Civita connection with is torsion-free. Thus the commutator of the covariant derivatives yields $[\nabla_\mu,\nabla_\nu]V^\rho = ...
3
votes
2answers
450 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 ...
3
votes
1answer
157 views

holomorphic sectional curvature and total scalar curvature

In a paper of Heier and Wong, It is written that from a pointwise argument due to Berger does follow that the scalar curvature (and thus also the total scalar curvature) of a Kaehler metric of ...
3
votes
1answer
154 views

Flattening a corner in a convex $d$-polytope (into $d-1$ dimensions, without overlap)?

I'm interested in the following question, which seems to be assumed all over the place (at least for 3 dimensions) in convex geometry, and which I cannot find a proof of. Suppose we have a corner ...
3
votes
2answers
162 views

Can one (block) diagonalize the curvature matrix of 2 forms on a Riemannian manifold?

Let $M$ be a smooth Riemannian manifold, let $R$ be the Riemannian curvature operator, and let $p$ be a point in the manifold. With respect to any orthonormal basis of the tangent bundle at the point ...
3
votes
1answer
238 views

geometric meaning of Ricci-flatness

What is the geometric meaning of Ricci-flatness? We know that if the Riemann tensor at a point vanished, manifold is flat at this point. but I don't know When the Ricci tensor vanished at a point, ...
3
votes
0answers
129 views

Strake's splitting theorem for complex sectional curvature

Strake's splitting theorem: Let $(M^n,g)$ be an open manifold with sectional curvature $K\ge 0$ and soul $\Sigma^k$. If the normal holonomy group of $\Sigma$ is trivial, $M$ splits isometrically into ...
2
votes
1answer
98 views

Compact Riemannian manifold with maximum average distance

Given a compact connected Riemannian $2$-manifold $M$ with positive curvature (thus by Gauß-Bonnet, $M$ is diffeomorphic to a 2-sphere) and diameter 1, what is the supremum (as $M$ varies over all ...
2
votes
1answer
213 views

Curvature of contour lines of a scalar field

How can I compute the curvature of the contour lines (equipotential lines) $\phi (\vec{r})=c$ for the scalar field $\phi (\vec{r})$ ? I expect the direction of the curvature vector to be along the ...
2
votes
1answer
173 views

curvature of curves in the space of gaussians measures

I have a sequence of symmetric positive definite matrices in $GL(n)$ (in my case, covariance matrices of some gaussians) and vectors $R^n$ (the mean of these gaussians). I consider this sequence to be ...
2
votes
2answers
197 views

Is there a combinatorial analogue of the Kazdan Warner theorem?

First let me state a result of Kazdan and Warner Let $M$ be a compact orientable two dimensional manifold. Let $f:M \rightarrow \mathbb{R}$ be a function that has the same sign as the Euler ...
2
votes
0answers
112 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 ...
1
vote
3answers
399 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.
1
vote
3answers
244 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 $p \in C$ ...
1
vote
0answers
207 views

Formula for the curvature of an induced connection

Let $\pi_P:P\to M$ be a principal $G$-bundle on a manifold $M$ endowed with a connection $A$ and $\pi_Q:Q\to M$ be a principal $H$-bundle on a manifold $M$. Let $f:P\to Q$ be a morphism of bundles ...
0
votes
1answer
138 views

volume of a submanifold implies bounds on curvature

I would like to ask the following question: Suppose an m-dimensional manifold in an n-dimensional euclidean space, choose some point on this manifold and take an n-dimensional ball of radius R centred ...
0
votes
1answer
205 views

curvature and volume growth

Let $M$ be a non-compact connected Riemannian manifold with $\mathrm{sec}_g=0$ and $\operatorname{vol} B(x,r)\geq c(n)r^n$ for any $r$, where $c(n)>0$. How to prove that $(M,g)$ is isometric to ...
0
votes
0answers
70 views

Exposition of the Calabi complex

I am interested in a complex derived by Eugenio Calabi in his article "On compact Riemannian manifolds with constant curvature". The complex is referenced as "Calabi complex" in various citing ...
0
votes
0answers
133 views

Integral of Square of Mean Curvature

Let us assume $H$ is the mean curvature of a compact surface in $E^3$ and $g$ is its genus. (1) When $g$ is arbitrary, we have $\int_{S^{2}}H^{2}dV=4\pi$ and $\int_{\Sigma}H^{2}dV\geq4\pi$. (2)When ...
0
votes
0answers
155 views

Value of convolution integral of Gaussian function and curvature of a circle segment

Is there an analytical expression for the following integral, assuming $\alpha\in(0,\pi)$ and $\sigma>0$? $$-\frac{1}{\sqrt{2\pi} \sigma} \int_{-\sin(\alpha/2)}^{\sin(\alpha/2)} \exp(-\tfrac12 ...
0
votes
0answers
94 views

General form of a metric affine connection with zero curvature

I have read somewhere that the general form of a metric affine connection whose Riemann curvature is zero is given by $$\Gamma^{i}_{ij}=A^{i}_{\alpha}\partial_{j}A^{\alpha}_{k}$$ where ...
0
votes
0answers
45 views

volume of a submanifold in a bounded region implies bounds on curvature

I would like to ask the following question: Suppose an $m$-dimensional submanifold in $\mathbf{R}^n$, such that there is a constant $l$, representing the largest number allowing an open normal bundle ...
0
votes
0answers
179 views

About the parallel transport and choice of connection

Thought Experiment Consider a 2-sphere, $S^2$, and let $p$ be a point at the equator. Case 1 Let us parallel transport a vector, $V$ from $p$ using the recipe: Move one unit of length East. Move ...
-4
votes
0answers
75 views

how to make Contravariant and Covariant tensors applicable to problems of curvatures in halfspace problems? [on hold]

Consider a material halfspace and assume it to be made of infinite number of layers of same material, such that when the material is loaded at the top surface, how to quantify the variation of ...