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3
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1answer
95 views

Special spheres: principal curvatures with different signs

For $\varepsilon > 0$, we say that a closed, connected and oriented immersed hypersurface $M^n$ of a riemannian manifold $(N^{n+1},g)$ is $\varepsilon$-convex whenever all principal curvatures of $...
4
votes
0answers
74 views

Positive and non-negative sectional curvature of semi-riemannian metrics

I am currently studying the techniques related to geometry of positive and non-negative sectional curvature of Riemannian metrics. In particular, I have done some work with Cheeger deformations. I ...
3
votes
1answer
94 views

Flatness in a neighborhood of a point condition

Suppose that we have a Riemannian Manifold $(M,g)$ whose curvature vanishes in an open neighborhood U of a point p. When does this imply that the metric is Flat ? In particular, does it happen ...
1
vote
1answer
481 views

How do we compute the even cohomology $H^{2i}(Q)$ of the affine hyperquadric?

Consider the affine hyperquadric $Q:=\biggl\{(z_1,...,z_{n+1})\in\mathbb{C}^{n+1}\biggl|\sum_{i=1}^{n+1}z_i^2=1\biggr\}\cong TS^n$. What is a reasonable Kähler metric for $Q$ (induced by the ...
1
vote
1answer
89 views

Comparison principle for viscosity solution

I am currently reading the paper "The Inverse Mean Curvature Flow and the Riemannian Penrose Inequality" written by Gerhard Huisken and Tom Ilmanen. https://projecteuclid.org/euclid.jdg/1090349447 I ...
2
votes
0answers
27 views

Small perimeter-minimizing disks on curved surfaces

Suppose that I have a smooth curved surface, and I choose an arbitrary point $Q$ on that surface. Say the Gaussian curvature at that point is $K$. What I am wondering is, is there an expression for ...
5
votes
2answers
348 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 ...
3
votes
1answer
130 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 ...
2
votes
2answers
419 views

A kind of “Curvature tensor” for higher dimensional tensors

I begin my question with a multilinear question then I will consider two local smooth analogies: Assume that $\alpha$ is a real valued symmetric $k$-tensor, that is a $k$-linear map $\alpha:\...
1
vote
1answer
147 views

holonomy of connection on gerbes

I am reading this notes of Hitchin to understand about gerbes. He defines gerbe by giving a collection of $2$ cocycles $g_{\alpha\beta\gamma}:U_\alpha\cap U_\beta\cap U_\gamma\rightarrow S^1$ with ...
4
votes
2answers
207 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 $M$ ...
6
votes
0answers
68 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 $(a,b)\in\omega^2\setminus\{(0,0)\}$ ...
1
vote
0answers
96 views

A reference for Poincaré's type inequality for vector fields

$\newcommand{\M}{\mathcal{M}}$ $\newcommand{\TM}{T\mathcal{M}}$ $\newcommand{\Ric}{\operatorname{Ric}}$ $\newcommand{\Volg}{\operatorname{Vol}_g}$ I would like to find a reference for the following ...
4
votes
1answer
257 views

Locally Riemannian Connection

Let $\Gamma^a{}_{bc}=\Gamma^a{}_{cb}$ be a symmetric connection whose curvature is $$R^a{}_{bcd}=\partial_c\Gamma^a{}_{bd}-\partial_d\Gamma^a{}_{bc}+\Gamma^a{}_{ec}\Gamma^e{}_{bd}-\Gamma^a{}_{ed}\...
2
votes
1answer
198 views

Second fundamental form and embeddings

Let $\Sigma$ be a smooth hypersurface of a $d$ dimensional smooth Riemannian manifold $(\mathcal M, G)$; we may see $G_x$ as a mapping from $T_x(\mathcal M)$ into $T_x^*(\mathcal M)$ so that $$ \...
12
votes
1answer
276 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 ...
2
votes
1answer
154 views

Fary-Milnor Theorem : Help following a proof on page 9

I am studying Fary-Milnor Theorem on total curvature of knots and I am stuck in a proof. He is proving on page 9: The Total curvature of a tame knot cannot equal the curvature of its type k(C) := ...
2
votes
0answers
169 views

Location of the endpoints of two parametric curves

I have two curves, $C_1$ and $C_2$ parametrized by $\theta$, the angle of the outward normal with the X-axis. $C_1$ is given by the following equations (say $r = 0.2$): \begin{align*} \frac{dx}{d\...
1
vote
0answers
133 views

Measure of distance between two curves described by their curvature

I have two curves both of which start at $(0,0)$. They satisfy the following equations: $ \begin{align*} \frac{dx}{d\theta} &= -\frac{\sin \theta}{\kappa(\theta)} \\ \frac{dy}{d\theta} &= \...
2
votes
0answers
74 views

How do conformal maps affect curvature?

Let $(\overline{M}^{n+1}, \langle \cdot, \cdot \rangle)$ be a riemannian manifold with riemannian connection $\overline{\nabla}$ and consider $M^n \subset \overline{M}$ an orientable hypersurface with ...
6
votes
0answers
118 views

Regarding a proof in the surgery theorem by Gromov and Lawson

I have a question regarding a proof in the article The classification of simply connected manifolds of positive scalar curvature written by Gromov and Lawson. The precise reference is: Gromov, ...
11
votes
1answer
404 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).
2
votes
2answers
135 views

Mean curvature upper bounds and area, or geodesic curvature upper bounds and length

Let $M$ be a closed manifold with non-torsion $\pi_2$, and $A$ a non-trivial free homotopy class of a map $f: S^2 \to M$. Let $S$ be the set of (immersed) class $A$ surfaces in $(M,g)$ with mean ...
3
votes
0answers
114 views

Integral of Gaussian curvature multiplied by mean curvature

Let $M$ be a 3-manifold with positive definite metric $g$, and let $S\subset M$ be an oriented 2-surface. For $x\in S$ let $K(x)$ be the Gaussian curvature and $H(x)$ be the mean curvature of $S$ at ...
2
votes
0answers
91 views

Surgery and Curvature on Foliation

Let $X$ be an oriented closed smooth $4$-manifold. Suppose that $TM$ admits a foliation $\mathcal F$ of dimension two, and admits a positvescalar curvature. Q: If we do the surgery on $X$ to reduce ...
9
votes
0answers
169 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, ...
3
votes
0answers
305 views

Curvature $\geq-1$ but not $\geq1$

(Edited again) In the following, for brevity, I will say that $$X\ \ \mathrm{has}\ \ \kappa_{\mathrm{max}}=k$$ if $X$ is a compact ($n$-dimensional with $n\geq2$, with empty boundary) Aleksandrov ...
1
vote
1answer
176 views

Under what condition the covariant derivative of Ricci operator along Killing vector field vanish?

Let $(M,g)$ be a Riemannian manifold and $V$ a unit Killing vector field on it. Under what condition on curvature tensor the following equation hold: $$\nabla_VQ=0,$$ where $Q$ is the Ricci operator ...
3
votes
0answers
135 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 ...
4
votes
0answers
339 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(...
6
votes
1answer
249 views

Reference for parallel transport around loop and its relation to curvature

It is a well known fact that the geometric meaning of a linear connection's curvature can be realized as the measure of a change in a fiber element as it is parallel transported along a closed loop. ...
1
vote
1answer
248 views

Riemannian Manifolds of Bounded Curvature

I am a complete newbie Riemannian Geometry with a particular application in mind so please excuse a lack of rigor in the question. Suppose I have a manifold with sectional curvature everywhere ...
2
votes
1answer
450 views

Mean curvature vector approximated for the discrete Laplace Beltrami Operator

"It is known that, for a curvature continuous surface, if the curve $\gamma$ shrinks to the point $v_i$, the (line) integral converges to the mean curvature $\kappa(v_i)$ of the surface at the point $...
2
votes
1answer
146 views

Relation of pseudo-torsion with curvature in degenerate plane

Question: I'd like to know if there is some reference or reasonable way to develop curve theory in a plane with degenerate metric $(\Bbb R^2, {\rm d}s^2 ={\rm d}x^2)$. Context: In Lorentz-Minkowski ...
1
vote
2answers
254 views

Derivation of the volume preserving mean curvature flow

~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ Picture above is from Huisken, Gerhard, The volume preserving ...
5
votes
3answers
214 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
0answers
110 views

Geometric interpretation of energy-momentum tensor and Lagrangian associated to a soliton equation

I have a question for you. BACKGROUND Consider an immersion $F\,:\;\mathscr S\;\longrightarrow\;\mathbb E^3$ of a surface $\mathscr S$ in the $3$-D euclidean plane $\mathbb E^3$ with canonical ...
16
votes
0answers
173 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 ...
10
votes
1answer
400 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 ...
14
votes
2answers
395 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 $f:M \to N$ ...
3
votes
1answer
787 views

A kind of surfaces

Is there a way* to prove that a Bonnet Surface $S$ in isothermal coordinates in $R^3$ with mean curvature ($H$) and Gaussian curvature ($K$) both non-constant and where $(H^2-K)=c$, (with c positive ...
6
votes
1answer
105 views

Limit of curvature near lightlike points

Let $\alpha \colon I \to \Bbb R^2_1$ be a regular curve and $t_0 \in I$ be such that $\alpha$ is lightlike at $t_0$, and not lightlike at $]t_0-r,t_0[$ for some $r>0$. Then, in that interval the ...
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vote
0answers
168 views

Surfaces in isothermal coordinates and particular PDE

From Brioschi's formula, if we have a surface in isothermal coordinates were $g_{ij}=E(u,v)*\delta_{ij}$ is the metric tensor, the gaussian curvature is: $K=-\frac{1}{2E}[\frac{\partial}{\partial u}(...
5
votes
0answers
224 views

“Correct” definition of signed curvature in Minkowski plane

We know that for $n\geq 2$ the de Sitter space $\mathbb{S}^n_1(r)$ and the hyperbolic space $\mathbb{H}^n(r)$ have constant curvature $1/r^2$ and $-1/r^2$, respectively. Looking at references such as ...
3
votes
0answers
83 views

Conformal Transformations that are Ricci Positive Invariant

Is there any known class of conformal transformations $\phi : M \to M$ of a riemannian/semi-riemanian manifold $(M,g)$ that have the property: $g$ is ricci-positive iff $\phi^* g$ ricci positive? ...
1
vote
1answer
79 views

Determining the rate of spread of geodesics when the sectional curvature is zero

I have posted this question in mathSE a few weeks ago (and proposed a bounty) but so far got no response. In the book Riemanian geometry (by do-Carmo), the following result is proved (Corollary 2.9 ...
0
votes
0answers
97 views

Bounds on Christoffel symbols

Let $\Omega$ be an open connected subdomain of a Riemannian manifold $M$ with boundary such that $\Omega$ is $\epsilon$ close to the boundary of $M$. Moreover, $\partial\Omega$ has two boundary ...
1
vote
0answers
71 views

Positive combinatorial curvature - upper bound for maximal number of vertices

$Thm.:$ Let $G$ be a finite graph embedded in a sphere with $n \geq 580$ vertices and $3 \leq deg(x) < \infty $ and everywhere positive combinatorial curvature $K_G(x) > 0$ for all $x \in V(G)$,...
5
votes
1answer
175 views

Evolution of $W_+$ and $W_-$ under the Ricci flow

In dimension $4$ the Weyl operator $W$ splits in two parts $$W_+:\Lambda^{2}_{+} \to \Lambda^{2}_{+}$$ and $$W_-:\Lambda^{2}_{-} \to \Lambda^{2}_{-}.$$ (a) Has there been a study of the evolution ...
2
votes
2answers
273 views

Curvature of singular Riemannian metric

Let $M$ be a differentiable manifold of dimension $n>1$ and $g$ a flat Riemannian metric on $M$. Consider $f:M\rightarrow \mathbb{R}^+$ a continuous function which doesn't have continuous first ...