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Questions tagged [curvature]

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1answer
132 views

determinant of curvature (notation issue)

This is when studying about Chern classes from Kobayashi and Nomizu. Let $\pi:E\rightarrow M$ be a complex vector bundle with fibre $\mathbb{C}^r$ and Group $G=GL(r,\mathbb{C})$. Let $p:P\rightarrow ...
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35 views

An integral estimate in conformal geometry

Let $g_0$ be a Riemannian metric of positive scalar curvature $R_0$ on a closed 4-manifold, and for some fixed constant $C$, define the set \begin{equation} \mathcal{S} = \{u\in C^\infty(M): ||u||_{W^{...
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51 views

Explicit parametrization of closed space curves of constant curvature

Joining arcs of helices it is pretty easy to obtain closed curves with constant curvature and $C^2$ regularity (see http://www.heldermann-verlag.de/jgg/jgg01_05/jgg0203.pdf). Joining arcs of Salkowski ...
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54 views

Geometrical regularity of the projection/normalization of a curve

Let $v:\mathbb{R} \rightarrow \mathbb{R}^3/{(0,0,0)} $ be a $C^\infty$ regular arc-length parametrization of a space curve. W.l.o.g. let us assume $v(0)=(1,0,0)$, $v'(0)=(1,0,0)$. Let $\bar{v}$ be ...
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0answers
132 views

A geometric rank of Riemannian manifolds

There are various ranks which have been assigned to a smooth manifold. The following ranks are two examples of such ranks: The maximum number of global independent vector fields which can be defined ...
3
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1answer
91 views

Sectional curvatures under simple maps

Suppose that we have a submanifold $X$ of $\mathbb{R}^n$ with the induced Euclidean metric, whose sectional curvatures we have a handle of (say, they are lower bounded by some $\kappa$). Is there a ...
4
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1answer
127 views

A variation on four-vertex theorem

Is it true that, if a closed, strictly convex curve has exactly four vertices (extrema of curvature), then any circle has at most four points of intersection with it?
5
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1answer
238 views

Principal curvatures of $\mathbb{R}^{n^2}$-embedded SO(n)

It's well known that the sectional curvatures of a Lie group, endowed with a left-invariant metric have a nice closed-form formula $k(X,Y) = \frac{1}{4} \|[X Y]\|^2$. I'm wondering if the following (...
3
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1answer
168 views

Realizing the cross product of $\mathbb{R}^3$ as the curvature tensor of a Riemannian metric on $\mathbb{R}^3$

Is there a Riemannian metric on $\mathbb{R}^3$ for which the corresponding curvature tensor $R$ satisfies $R(X,Y)Z=(X\wedge Y)\wedge Z$? I have already discussed this question in the following post ...
3
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1answer
119 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
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77 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
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1answer
98 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 ...
0
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1answer
487 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
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1answer
92 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 ...
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0answers
31 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
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2answers
359 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
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1answer
137 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 ...
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2answers
431 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:\...
2
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2answers
188 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
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2answers
214 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 $M$ ...
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0answers
72 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 $(a,b)\in\omega^2\setminus\{(0,0)\}$ ...
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0answers
98 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
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1answer
260 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
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1answer
229 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
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1answer
291 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
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1answer
171 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
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0answers
171 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\...
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0answers
141 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
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0answers
80 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 ...
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0answers
127 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
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1answer
422 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
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2answers
141 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
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0answers
126 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
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0answers
97 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 ...
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179 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, ...
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0answers
306 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
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1answer
204 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
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0answers
138 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 ...
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363 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
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1answer
302 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
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1answer
260 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
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1answer
491 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
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1answer
150 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 ...
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2answers
269 views

Derivation of the volume preserving mean curvature flow

~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ Picture above is from Huisken, Gerhard, The volume preserving ...
5
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3answers
226 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 ...
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0answers
117 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 ...
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0answers
190 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
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1answer
431 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 ...
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2answers
399 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
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1answer
790 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 ...