Questions tagged [dg.differential-geometry]
Complex, contact, Riemannian, pseudo-Riemannian and Finsler geometry, relativity, gauge theory, global analysis.
8,645
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Is C^{k+1}(X) compactly contained in C^{k}(X) for a closed manifold X?
Hi all,
I apologize if this question is too low level for mathoverflow. I'm happy to move it to math.stackexchange if so.
Let $X$ be a closed manifold, let $k$ be a nonnegative integer and let $C^k(...
2
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2
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869
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How else can we describe the volume of a lagrangian submanifold in a Kahler manifold?
Suppose $(V^{2g}, g, \omega, J)$ is an almost Kahler manifold. ie. $(V,\omega)$ is a symplectic manifold with $\omega$-compatible almost complex structure $J$ ($J$ is a symplectomorphism) and such ...
2
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1
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537
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Quermassintegrals as mean curvature integrals
It is well-known that the quermassintegrals $W_{i}$ of a convex body $K\subset \mathbb{R}^{n}$ can be written as a mean curvature integral
$$ W_{i}(K)=n^{-1}\int_{\partial K}H_{i-1} d\mathcal{H}^{n-1},...
2
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1
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307
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Connecting tangents of convex curves: at some point orthogonal?
Let $a(t)$ and $b(t)$ be two smooth, nested convex curves in the plane, $t\in[0,1]$:
Suppose the parametrization of $a()$ and $b()$ is such that $\dot{a}(t)$ is ...
2
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1
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265
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Riemannian Hausdorff distance between two conjugacy classes in a compact Lie group
I am interested in the distance between two conjugacy classes in a group like $SO(n)$. However let's consider $U(n)$ for simplicity. My conjecture is that the Hausdorff distance between the conjugacy ...
2
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1
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189
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if $N$ is a submanifold of $M$, then what is the relation between $J^1 N$ and $J^1 M$?
I was asking myself if there exists a sort of canonical relation between the standard contact structure on $J^1 N$ and $J^1 M$, for an arbitrary submanifold $N$ of $M$.
My starting point is that, ...
2
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434
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Density of holomorphic sections
Hello!
I am reading an article in which there is the following statement:
Let $E\rightarrow X$ be a holomorphic vector bundle. The holomorphic sections of $E$ over a coordinate neighbourhood of $X$ ...
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452
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important vector bundles
I was wondering if any vector bundles on a manifold other than the tangent bundle give topological invariants. I guess stiefel Whitney classes also come from the inverse bundle - but other than that.
2
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310
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Seiberg-Witten equation on S^2\times S^1
What are the irreducible solutions of Seiberg-Witten equation on S^2\times S^1?
Thanks.
2
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2
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610
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Homeomorphism between the boundary of the Poincare disc S1 and its Gromov Boundary
Given the Poincare Disc $D$ and its ideal boundary $S^{1}$, I want to construct a homeomorphism between $S^{1}$ and the gromov boundary of $D$, $\partial D$, of equivalence classes of geodesics given ...
2
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1k
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Harmonic coordinates on Riemannian manifolds
I'm trying to read the paper of Jost and Karcher on the existence of harmonic coordinates on a ball whose size only depend on the injectivity radius and a two sided bound on the curvature.
...
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423
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Homology and submanifolds...
I'm reading a paper which includes the following line, and I can't find a reference anywhere to the result the authors mention:
"Let M be a compact orientable embedded minimal hypersurface of a ...
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2
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645
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Varieties, Frechet Completions, and Regular Functions
Take an algebraic variety $V$, and its set of smooth functions $C^{\infty}(V)$. One can endow $C^{\infty}(V)$ with a canonical locally convex topology (the seminorms are defined using the local ...
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807
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Frobenius Theorem
Say a manifold M has 3 vector fields S,T and R whose Lie brackets satisfy the equations $[S,T]=R$, $[R,S]=T$ and $[T,R]=S$
Then I suppose the following properties hold for M,
There exists a metric ...
2
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150
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Gluing local holomorphic connections
On page $180$ of Complex Geometry by Daniel Huybrecths, he defines the so called Atiyah class of a holomorphic vector bundle by the Čech cocycle $$A(E)=\{U_{ij}, \psi^{-1}_j \circ (\psi^{-1}_{ij}d\...
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For a closed Riemannian manifold $M$, must the set of points with non-unique closest points to a closed submanifold $S$ of $M$ be of 0 volume measure?
Let $(M,g)$ be a closed (compact without boundary) Riemannian manifold of finite dimension, with the volume measure $\mu:= \mu(E):=\int_{E}dvol_g \forall E \in \mathcal{B}(M),$ the Borel sigma algebra ...
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Does the group of compactly supported diffeomorphisms have the homotopy type of a CW complex?
It is known that the group of diffeomorphisms of a compact manifold with the natural $C^{\infty}$ topology has the homotopy type of a countable CW complex. See for instance this thread: Is the space ...
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150
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Osculating sphere at point of maximal curvature lies to one side
I'm looking for the higher dimensional version of this post, which says that given a curve $\gamma \subseteq \mathbb{R}^2$, the osculating circle will lie to one side of the curve at points of maximal ...
2
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1
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Why is this subset associated to a $2$-tensor dense?
Let $S$ be a symmetric $(0, 2)$ tensor on a Riemannian manifold $M$. Define $E_S : M \to \mathbb{Z}$ by $E_S(x) = \left(\text{the number of distinct eigenvalues of } S_x\right)$. I've seen the ...
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240
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Make a Riemannian metric real analytic in some coordinates
Suppose I have a smooth (high dimensional) Riemannian manifold, and fix a base point $x_0$. Can I pick a coordinate system near $x_0$ such that the metric is real analytic in some of the variables (...
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216
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Existence of diffeomorphism interpolating affine map and identity
$\newcommand{\R}{\mathbb{R}}$Suppose $\Omega$ is a bounded, convex domain in $\R^{m}$. Fix $x_1, x_2\in\Omega$ and an invertible matrix $A\in\mathrm{GL}^{+}(m)$ with positive determinant.
Let $U\...
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265
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If there exists a function on a Riemannian manifold such that its Hessian matrix is the identity matrix?
In Euclidean space $\mathbb{R}^n$, $n\geq 2$, the Hessian matrix of the function $\frac{|x|^2}{2}$ is the identity matrix. While on a smooth manifold $(M^n, g)$, do there exists a function on $(M^n, g)...
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132
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Upper bound on volume growth of area minimizers
Let $M^n$ be a complete simply connected Riemannian manifold with $\operatorname{sec}_M \leq 0$ (i.e. a Hadamard manifold) and assume that there is a constant $a \geq 0$ such that $\operatorname{sec}...
2
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275
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Volume of submanifold as integral of delta-function
Let $M$ be an $n-m$ dimensional sub-manifold of $\mathbf R^n$ defined by the following set of equations:
\begin{equation}
f_1(\vec x)=0, \\
\vdots \\
f_m(\vec x)=0,
\end{equation}
(where $\vec x$ are ...
2
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196
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On a closed manifold whose curvature is close to "hyperbolic"
$\DeclareMathOperator\Vol{Vol}$In 1.7 on p.224 of the following paper, there is a rigidity result for compact manifolds whose sectional curvature is almost $-1$.
Gromov, M.. Manifolds of negative ...
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297
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Determine the form of the wave equation in Minkowski space on the line element $ds^2 = -dt^2 + a(t)(dx^2 + dy^2 + dz^2)$
The wave equation in Minkowski space can be given as
$-\frac{\partial^2\phi}{\partial t^2}+\frac{\partial^2\phi}{\partial x^2}+\frac{\partial^2\phi}{\partial y^2}+\frac{\partial^2\phi}{\partial z^2}= ...
2
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157
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Are vector bundles acyclic for $\Gamma_c$?
Let $X$ be a paracompact topological space or a manifold (which is not a particular case since the structure sheaves are different). It is well-known that vector bundles (more generally, $\mathcal{O}...
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140
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Morse approximation with bounded number of critical points
Let $(M^3,g)$ be a compact Riemannian 3-manifold and let $f\in C^{\infty}(M)$ be a smooth function. Does there exist a constant $k>0$ (possibly depending on $M$ and $g$) such that $f$ can be $C^2$-...
2
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149
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Hyperboloids in Minkowski geometry
Let $(\mathbb R^{1+2},\eta)$ be Minkowski with the metric $\eta= -dt^2+(dx^1)^2+(dx^2)^2$. Suppose $\Sigma$ is a smooth timelike hypersurface and denote by $h$ the second fundamental form on $\Sigma$. ...
2
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111
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Quasiconformal map from a subset of $\mathbb{C}$ to a polytope
Question. Does a quasiconformal map exist between a subset of $\mathbb{C}$ (such
as a unit disc or rectangle) and a polytope?
Here, I take a polytope to be a two-dimensional surface that could be ...
2
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1
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362
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Locally free sheaves and vector bundles over smooth connected projective curve
Let $X$ be a connected smooth projective curve over an algebraically closed field $K$. Let $\mathcal{F}$ be a locally free sheaf on $X$ and $\mathcal{E}$ a subsheaf of $\mathcal{F}$, which is again ...
2
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1
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128
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Infinitely many deformation equivalent Hodge diamonds
Let $S$ be a connected open complex manifold. Let $\phi:X\to S$ be a proper holomorphic submersion with connected fibers. Can the fibers of $\phi$ have infinitely many distinct Hodge diamonds?
An ...
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1
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217
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To what extent is a vector bundle on a smooth manifold determined by its restriction to the complement of a closed smooth submanifold?
The question is a follow-up to this one.
Let $N\subset M$ be a closed smooth submanifold of codimension $k\geq 1$ of a smooth manifold $M$. Let $E_1\rightarrow M$ and $E_2\rightarrow M$ be two vector ...
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277
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Restriction of diffeomorphisms homotopic to identity to the boundary
Let $M$ be a smooth manifold with boundary $\partial M$. Let $Diff_0(M)$ be the group of all diffeomorphisms homotopic to identity. According to this article (Page 6, section "
Beyond mapping class ...
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2
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393
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Reconciling some result about the exponential map, the Chow-Rashevskii theorem, and $\mathrm{Diff}_0(M)$
Let $M$ be a $C^{\infty}$ manifold $C^{\infty}$-diffeomorphic to $\mathbb{R}^d$. I've recently come across some results which I'm trying to reconcile. Let $\mathfrak{X}(M)$ denote the set of ...
2
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1
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110
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Is it possible to tessellate a torus minus a disk using hyperbolic right-angled pentagons?
I am trying to construct a compact hyperbolic surface tessellated with hyperbolic right-angled polygons with $n \ge 5 $ edges. I found quite easily a way to do it for $n$ even, but the odd case seems ...
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339
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Applications of Generalized Geometry to Theoretical Physics [closed]
I'm looking for some topics on Generalized Geometry applied to Physics for a master thesis. I took an introductory course last year, and I have a degree in both Mathematics and Physics. I would ...
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309
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Totally geodesic submanifolds of bi-invariant Lie groups
Let $G$ be a Lie group equipped with a bi-invariant metric. I have some questions concerning the totally geodesic and the flat (all sectional curvatures zero) submanifolds of $G$.
I known that every ...
2
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1
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451
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Gaussian null coordinates
I find it hard to find information on the so-called "Gaussian null coordinates", which Wikipedia says is used to describe "near horizon geometries". Can someone provide a reference where I can read ...
2
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1
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512
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Dirichlet problem for manifold, how to prove $W^{1,2}_0(\Omega)$ solution is $C^{2,\alpha}(\bar{\Omega})$?
Let $M$ be an n dimensional Riemannian manifold without boundary. Let $\Omega \subset M$ be a bounded domain with smooth boundary. Let $f \in C^{\alpha}(\bar{\Omega})$, Consider the Dirichlet problem.
...
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237
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What is the Weak Maximum Principle for Scalars and how is it Derived?
I am currently reading 'Lectures on Ricci Flow' by Peter Topping and I have got to Chapter 3 where he states the 'weak maximum principle for scalars'. Suppose for $t \in [0,T]$ for finite $T$ that $g(...
2
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1
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328
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Poisson equation on noncompact manifold
Let $(M,g)$ be a complete non-compact manifold with bounded geometry, such that the Sobolev embeddings hold and $C^\infty_c$-functions are dense in $L^p_k$ space.
For the equation
$$\Delta u=f,$$
...
2
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2
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346
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Einstein warped product manifold Ricci flat
Let $(M,g)=(N,\ddot{g})\times f(B,\bar{g})$ be an Einstein warped-product manifold Ricci flat (i.e. $Ric=\lambda g$ with $\lambda=0$) where $f:N \rightarrow (0, \infty)$ (positive scalar function) and ...
2
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320
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Is the conformal compactification of a convex-cocompact hyperbolic manifold $M$ conformally diffeomorphic to the convex core of $M$?
I am not a hyperbolic geometer, so I apologize if I get anything wrong here, and please correct me.
The conformal compactification $\overline {\mathbb{H}^n}$ of hyperbolic $n$-space $\mathbb{H}^n$ ...
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549
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Confusion in definition of Gerbes in Hitchin's notes
I am reading Nigel Hitchin's notes to understand about gerbes.
It starts the article by saying the following :
Before giving a definition, it’s worthwhile to recognize when we, as
mathematicians, ...
2
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2
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485
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A strongly non-integrable distribution
What is an example of a three-dimensional smooth distribution $D$ of $\mathbb{R}^4$ with this property:
Not only $D$ is not integrable but also there is no a two-dimensional ...
2
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1
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246
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Is Bregman divergence independent of coordinates?
Question
Is Bregman divergence free of coordinates?
Although it is invariant w.r.t. which local affine coordinate you take, is it possible to prove that it does not change w.r.t. an arbitrary change ...
2
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1
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140
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$2$ dimensional foliations of space whose leaves contain the trajectories of a given vector field
Assume that $X$ is a non-vanishing vector field on $\mathbb{R}^3$.
Is there a $2$-dimensional foliation of space such that every trajectory of $X$ is contained in a leaf of the $2$-dimensional ...
2
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2
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143
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Intersection of segments on a plane
Given a set A composed by six non collinear distinct points on the plane, let us consider only the partitions whose elements are pairs of points in A. Then, we call the set of such partitions by P(A). ...
2
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1
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405
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The Lie algebra of gradient vector fields
Assume that $M$ is a differentiable manifold. Is there a Riemannian metric on $M$ such that the space of all gradient vector fields on $M$ would be closed under the Lie bracket?