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28
votes
2answers
2k views

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, ...
11
votes
1answer
377 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 ...
10
votes
3answers
1k 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 ...
10
votes
2answers
386 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 ...
9
votes
1answer
429 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
683 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 ...
8
votes
4answers
3k 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 ...
8
votes
3answers
2k 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 ...
7
votes
2answers
654 views

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 ...
6
votes
1answer
358 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 ...
6
votes
1answer
192 views

Compact Eucledean hypersurfaces with “almost” constant H_k curvature

Let $M$ be an Eucledean $n$-dimensional compact hypersurface with constant $H_k$ curvature, where $k=1,...n$. A theorem by A.Ros tell us that so $M$ is an Eucledean sphere. Does anybody know if there ...
6
votes
1answer
229 views

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

If we consider $G$ a Lie group with left invariant riemannian metric its sectional curvature is nonnegative, when this metric is positive? I thought a little about and only found $SU(2)=S³$. In ...
6
votes
0answers
234 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
2k 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
90 views

Convex embedding with a positivity condition

We have a $n$-dimensional hypersurface $\Sigma$ embedded in the Euclidean $(n+1)$-space $\mathbb{R}^{n+1}$. We know that $\Sigma$ is compact without boundary, convex (not necessarly strictly convex), ...
5
votes
1answer
713 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 ...
5
votes
0answers
234 views

On the curvature tensor with certain conditions

Let $(M^{n+m},g)$ be a Riemannian manifold and let $\lbrace X_1,...,X_n,Y_1,...,Y_m\rbrace $ be a locally orthonormal frame for $M$($3\leq n,m$). If we suppose the curvature tensor $R$ of $g$ ...
4
votes
5answers
571 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 ...
4
votes
2answers
214 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 ...
4
votes
3answers
765 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.
4
votes
2answers
202 views

Holonomy of a Ricci-flat affine connection

There is some link between Ricci-flatness and reduction of holonomy. For example a Kahler manifold is Ricci-flat if and only if it has at most $SU(n)$ holonomy rather than $U(n)$, and it's apparently ...
4
votes
1answer
180 views

Gaussian Curvature of Exponentiated 2-Planes

Consider a Riemannian manifold $M$ with sectional curvatures $K\ge 0$ and let $\Pi$ be a 2-plane in the tangent space of $M$ at a point $p$. In a small enough neighborhood $U$ of 0 the exponential map ...
4
votes
0answers
83 views

Flows associated with Killing fields

Let $M$ be a Riemann manifold and $p, q$ two points on a geodesic $\sigma$ which are isotropically conjugate. That is, there is a Jacobi field along $\sigma$ vanishing at $p$ and $q$ which is the ...
4
votes
0answers
147 views

Obtaining the metric from the mixed Ricci tensor $R^i{}_j$

In chapter 5 of the book "Einstein Manifolds", Arthur Besse discusses the possibility to find the metric $g$ when knowing the Ricci curvature tensor $Ric(g)$ ($=R_{ij}$). But what do we know about ...
4
votes
0answers
187 views

Manifolds with a lower degree of regularity

I've been reading a paper about regularity theory for a P.D.E in a non-smooth domain(see the reference below). There, the authors consider domains of $R^n$ with regularity of class $W^2 ...
3
votes
2answers
1k 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
1answer
109 views

The sign of the mean curvature on convex cones in three dimensions

my question is as follows. It is known that a closed smooth curve in $R^2$ is convex iff its (signed) curvature has a constant sign. I wonder if one can characterize smooth convex cones in $R^3$ in a ...
3
votes
1answer
257 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
2answers
1k 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
177 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
319 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
346 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
42 views

Compressing a hypersurface on the sphere

Let $M^n$ be a compact, connected, orientable hypersurface of the unit sphere $S^{n+1} \subset \mathbb{R}^{n+2}$. Suppose $M$ is contained in the northern hemisphere $S_+^{n+1}$ and has nonzero ...
3
votes
0answers
87 views

Growth of norm of curvature under direct sum or existence of universal connection

For a Hermitian vector bundle E over a Riemannian $(X,g)$ and a unitary connection A we may define the sup norm of the curvature by: $$\|R_A\|=\sup \{\| tracefree (R_A (v))\|_{op} \mid v \in \Lambda ...
3
votes
0answers
76 views

Classification of manifolds with ${\rm Ric}\geq 0$ wrt fundamental group

Note that $n$-manifolds $M$ with ${\rm Ric}\geq 0$ has a fundamental group of polynomial growth of degree $\leq n$ (proof : use Bishop volume theorem). (Here a group $\Gamma$ is said to have ...
3
votes
0answers
152 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
177 views

Symmetries of non-Riemannian curvature tensor

The curvature tensor, $R_{ab}{}^c{}_d$, can be obtained from a connection which not necessarily is a metric connection. By construction it is antisymmetric in the first two indices, since roughly ...
2
votes
1answer
128 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
320 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 ...
2
votes
1answer
635 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
135 views

Ricci Curvature and the Chern Class of the Levi-Civita

For a (compact) Kahler manifold $M$, the Ricci tensor is the symmetric $2$-form $$ r(u,v) = \text{tr}\big( w \mapsto (D_wD_u - D_uD_w - D_{[u,w]})v\big). $$ The Ricci curvature is the $2$-form $$ ...
2
votes
1answer
110 views

Regularity - mean curvature equation

In my research I arrived at the following equation: $$ \int_B \frac{\nabla u \cdot \nabla \varphi}{ \sqrt{1+|\nabla u|^2}}=\int_B f \varphi, (*)$$ for every $\varphi \in C^1(B)$, which is a weak form ...
2
votes
1answer
122 views

The Laplacian of an expression involving the Ricci tensor

While doing some computations on a compact Riemannian manifold I have reached the following expression: $$ \Delta_y \big( Ric_y (\exp_y ^{-1} x, \exp_y ^{-1} x) \big) (x)$$ where $\Delta_y$ is the ...
2
votes
1answer
115 views

Limited expansion of mean curvature of geodesic spheres

I am working with the Laplacian on a Riemannian manifold $(M,g)$ (compact, without boundary). In spherical geodesic coordinates $(r, \sigma)$ around some arbitrary $x \in M$ (where $\sigma$ denotes ...
2
votes
1answer
196 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
1answer
97 views

Prescribing an induced metric

We know that, if we have a surface $z=f(x,y)$ with Euclidean space being ambient manifold, the induced metric is as follows (in matrix form): $$g=\begin{bmatrix} 1+\left ( \frac{\partial ...
2
votes
2answers
361 views

Calculating the Riemann Christoffel tensor for a diagonal metric

I am trying to calculate the entries of the Riemann curvature tensor $R^m_{\phantom{m}ijk}$ for the metric $g_{ij}$. The Riemann-Christoffel tensor is given as \begin{align} R^m_{\phantom{m}ijk} = ...
2
votes
1answer
173 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 ...
2
votes
2answers
239 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
75 views

Bounding distance between geodesics in manifolds with nonpositive curvature

This is a duplicate of a question at the stackexchange which was not answered. I've recently read (in some notes by Mark Pollicott) the following related claims, which, although quite intuitive, I ...