Questions tagged [curvature]
Gaussian curvature, mean curvature, sectional curvature, scalar curvature, curvature tensors (Riemann, Ricci, Weyl)
29
questions
44
votes
2
answers
10k
views
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 ...
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 $...
17
votes
2
answers
5k
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 ...
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:$$\begin{cases} x'=P(...
13
votes
4
answers
6k
views
When is a Riemannian metric equivalent to the flat metric on $\mathbb R^n$?
I'm looking for an easily-checked, local condition on an $n$-dimensional Riemannian manifold to determine whether small neighborhoods are isometric to neighborhoods in $\mathbb R^n$. For example, for ...
3
votes
0
answers
161
views
Flat Riemannian metrics adapted to quadratic vector fields with center
Assume that $P(x,y),Q(x,y)\in \mathbb{R}[x,y]$ are two polynomials of degree $2$ with $P(0,0)=Q(0,0)=0.$
Suppose that the vector field $$\begin{cases} x'=P(x,y)\\ y'=Q(x,y) \end{cases}$$ has a center ...
20
votes
3
answers
8k
views
Curvature of a Lie group
Since a lie group is a manifold with the structure of a continuous group, then each point of the manifold [Edit: provided we fix a metric, for example an invariant or bi-invariant one] has some scalar ...
12
votes
1
answer
625
views
When is the hull of a space curve composed of developable patches?
Let $C$ be a smooth curve in $\mathbb{R}^3$ that lies entirely on its convex hull,
$\cal{H}(C)$.
Under what conditions on $C$ is $\cal{H}(C)$ the union of developable surface patches?
I believe ...
3
votes
1
answer
4k
views
Relationship between sectional curvature, bisectional curvature and conjugate points
Tian has defined bisectional curvature for unit and perpendicular tangent vectors $X,Y$ as follow
$$R(X,Y,X,Y)+R(X,JY,X,JY).$$
If bisectional curvature be constant, is there any relationship between ...
1
vote
1
answer
175
views
Conditions for Lipschitzness of boundary normal vector, almost everywhere
Let $C$ be a nonempty closed subset of $\mathbb R^n$. It is known that any such set satisfies the following condition
(Unique CPP a.e). For almost every $x \in \mathbb R^n$, there exists a unique ...
1
vote
3
answers
554
views
Surface analog of clothoid: curvatures covering $\mathbb{R}$
The clothoid $C$, a.k.a. the Euler spiral,
is one among many curves with
the property that its curvatures cover $\mathbb{R}$
in the sense that, for every $x \in \mathbb{R}$,
there is a point $p \in C$ ...
35
votes
10
answers
8k
views
Some questions about scalar curvature
Recall that the scalar curvature of a Riemannian manifold is given by the trace of the Ricci curvature tensor. I will now summarize everything that I know about scalar curvature in three sentences:
...
34
votes
7
answers
4k
views
What is the best way to draw curvature?
This is more of a pedagogical question rather than a strictly mathematical one, but I would like to find good ways to visually depict the notion of curvature. It would be preferable to have pictures ...
17
votes
2
answers
1k
views
Are there some intrinsic invariants of surfaces other than Gaussian curvature?
The principal curvatures of a surface is denoted by $\kappa_{1}, \kappa_{2}$.
Let $P(x,y)$ be a polynomial with real coefficients. Assume that $P(\kappa_{1}, \kappa_{2})$ is an intrinsically ...
10
votes
2
answers
899
views
Techniques to solve a non-linear differential equation related to curvature
Many years ago, I considered the following non-linear differential equation:
$y=y''\cdot(1+y'^{2})^{-3/2}$
This equation expresses the equality between the value of a given function $y\in C^{2}(R)$ ...
8
votes
5
answers
3k
views
Variation of curvature with respect to immersion?
Let $M$ be a smooth surface and let $f: M \to \mathbb{R}^3$ be a family of immersions given by
$$ f(t) = f_0 + tuN_0, $$
where $f_0$ is some initial immersion, $N_0$ is the associated Gauss map, and ...
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 ...
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 ...
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 ...
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 ...
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 ...
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 (...
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?...
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$ ...
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 ...
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$$ where $H$ is the mean curvature (defined w.r.t. a choice of continuous unit normals on $S$.) Is there a ...
3
votes
1
answer
578
views
Upper bound on the sectional curvature of the orthogonal group
Consider the orthogonal group $O(n)$ as a Riemannian manifold endowed with the usual (bi-invariant) metric $\langle P, Q \rangle_A = \textrm{Tr}\ P^\top Q$ for tangent vectors $P, Q$, with $$T_A O(n) =...
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 ...
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 ...