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Questions tagged [dg.differential-geometry]

Complex, contact, Riemannian, pseudo-Riemannian and Finsler geometry, relativity, gauge theory, global analysis.

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Characterisation of epimorphisms in the category of differentiable stacks

Let $\mathcal{D},\mathcal{C}$ be stacks and $F:\mathcal{D}\rightarrow \mathcal{C}$ be a morphism of stacks. Further assume these $\mathcal{D},\mathcal{C}$ are differentiable stacks, that is there ...
5
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1answer
112 views

Is every Lie subgroup of a Lie group isometric to all its conjugates?

Let $G$ be a Lie group with a left invariant metric. Assume that $N$ is a Lie subgroup of $G$. For a given $g\in G$, are $N$ and $g^{-1} N g$ necessarily isometric Riemannian manifold when they ...
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0answers
33 views

Terminology for a foliation that is only tangentially smooth

I'd like to get some information and references, starting with a name, for the following quite common situation, for a smooth (i.e. $C^\infty$) $n$-manifold $M$. A partition $\mathcal{L}$ of $M$ is ...
6
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1answer
135 views

Stack associated to Lie group and manifold

Given a Lie group $G$, we have a Lie groupoid $(G\rightrightarrows *)$ and stack $BG=B\mathcal{G}$ of principal $G$ bundles. Given a smooth manifold $M$, we have Lie groupoid $(M\rightrightarrows M)...
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37 views

Codinate geometry [on hold]

Calculate the lenght between the two points X(1/2,1/2) and Y(-1/2,-1/2) What is the value of R, if the distances between the two point (4,2) and (1,r) is 3 units.
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1answer
95 views

Vanishing product of a closed and coclosed form on a Riemannian manifold

For a (compact) Riemannian manifold $(M,g)$, can it happen that for a non-zero form $\text{d}^*\omega$, and a smooth function $f$ such that $\text{d}f \neq 0$, we can have $$ \text{d}f \wedge \text{d}^...
3
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1answer
85 views

Globally defined integral curves on the tangent bundle

Let $(M,g)$ be a riemannian manifold and $TM$ its tangent bundle. We know that if, for instance, $M$ is compact, any integral curve of any vector field on $M$ can be defined in the whole $R$ and not ...
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101 views

Constructing new complex manifolds out of old

It is not difficult to build new manifolds out of old in the smooth category, for example taking the direct product or constructing a fiber bundle, taking the level set of a regular value of a smooth ...
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0answers
42 views

Relating the components of the Riemann curvature tensor to the second partials of the components of the metric

I am currently reading through a proof of Proposition 6 in Chernoff's theorem and discrete time approximations of Brownian motion on manifolds OG Smolyanov, H Weizsäcker, O Wittich - Potential ...
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0answers
31 views

I want to obtain 'r' from the equation second time derivative of r is equal to a constant divided by square of r plus a constant? [on hold]

The solution expression should be like 'r=....', where the right hand side of the solution expression should not contain r. what to do? please help.
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0answers
46 views

Holonomic sections $C^\infty(M)$-generate jet bundle

Given a vector bundle $E \to M$ with a corresponding $k$-th jet bundle $J^kE \to M$, denote by $j^k : \Gamma(E) \to \Gamma(J^kE)$ the $k$-th jet prolongation $(k \in \mathbb{N} \cup \{0\})$ and recall ...
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56 views

Associated bundle construction and classifying space

Let $\theta:G\rightarrow H$ be a morphism of Lie groups. Given $G$ we have classifying space $BG$ and given $H$ we have classifying space $BH$. This $\theta:G\rightarrow H$ gives a map $B\theta:BG\...
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0answers
127 views

Invariant polynomials in curvature tensor vs. characteristic classes

Let $M$ be an $4m$-dimensional Riemannian manifold. We can then form the Pontryagin classes $p_k(TM)$ of the tangent bundle using Chern-Weil theory. For any sequence of numbers $k_1, \dots, k_l$ such ...
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0answers
36 views

Estimate on Covariant Derivatives of Coordinate Derivatives

I am currently reading Topping's lecture notes on Ricci flow. At one point in the narrative (page 65) he says that using the fact that $\bigg(\frac{\partial}{\partial t}\nabla - \nabla \frac{\...
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2answers
223 views

Compact complex affine Kähler manifold is a torus

Before giving a motivation let me ask the precise question firstly. By a complex affine manifold I mean a complex manifold $M$ with the property that there exists an holomorphic atlas for which ...
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0answers
67 views

Global solution of second order ODE defined on riemannian manifold

Consider the differential equation $\nabla \dot X + \frac{3}{t} \dot X + gradf(X) =0$, defined on a riemannian manifold $(M,g)$ ($ \nabla$ is the Levi-Civita connection and $gradf(X)$ is the ...
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1answer
106 views

A totally geodesic triangulation

Let $M$ be a compact orientable $n$ dimensional Riemannian manifold. Is there a triangulation of $M$ such that every $k$ dimensional face of each simplex is a totally geodesic submanifold, $\forall k ...
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0answers
55 views

Joining metrics of positive Ricci curvature

Let $M$ be a smooth manifold such that there is a closed submanifold $S\subset M$ with a Riemannian metric $g_S$ given by the restriction of a Riemannian metric on $M$ satisfying $\mathrm{Ricci}_{g_S}(...
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0answers
139 views

Completeness is a conformal invariant

In this article about completeness of semi-riemannian manifolds the author writes (in section 2.2) that it is unkown if the following statement holds: A compact indefinite manifold which is conformal ...
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0answers
46 views

Find wrapping angle of helix on a torus

I need some help in calculating the wrapping angle of a spiral helix wrapped on a torus with constant angle against all the meridians of the torus. The wrapping angle (or the angle measured around and/...
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1answer
117 views

Principal Symbol for the Ricci-DeTurck Flow

I am following some lecture notes on Ricci flow and reached the section where we linearize the Ricci tensor and obtain the principal symbol for the resulting operator. We have $T \in \: \Gamma(Sym^2 ...
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0answers
76 views

Requirement for weak pullback to be a Lie groupoid (Moerdijk)

Let $\phi:\mathcal{G}\rightarrow \mathcal{K}$ and $\psi:\mathcal{H}\rightarrow \mathcal{K}$ be morphisms of Lie groupoids. Is it necessary for $\phi_1:\mathcal{G}_1\rightarrow \mathcal{H}_1, \...
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0answers
17 views

Find local coordinates of end points of a time varying line (w.r.t. two other objects) to minimize line-length variance

Line A'A rotates cyclically around point A' in a fixed time varying pattern and coordinates of the point A' are fixed (i.e. line A'A has only one degree of freedom). Line BB' can move along x-axis and ...
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0answers
121 views

Taylor Expansion on a Riemannian Manifold in Normal Coordinates

Let $\phi: (M,g)\hookrightarrow (N,\tilde{g})$ be an isometric embedding of a Riemannian manifold $M$ of dimension $m$ into a Riemannian manifold $N$ of dimension $n$. I am interested in trying to do ...
2
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1answer
97 views

$(M,g)$ is complete iff $(\tilde{M},\tilde{g})$ is complete (non-Riemannian version)

I'm not sure if this question is too low level for Math Overflow (so feel free to move this to SE if you think it is). Inspired by this and this question I'm wondering if the following statement is ...
3
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0answers
94 views

A concept weaker than geodesibility of flows which is possibly useful in limit cycle theory

The main objective of this post is to apply the Gauss Bonnet Theorem to count the number of limit cycles of a polynomial vector field as described in this MO post and its linked MO posts But in this ...
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1answer
82 views

Requirement that source and target maps are surjective submersions

Definition I am aware of for Lie groupoid is that (among other things) the source and target maps $s,t:\mathcal{G}_1\rightarrow \mathcal{G}_0$ are submersions. On page 9 of Du Li's thesis Higher ...
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0answers
227 views

Geometric bang-bang theorem for nonlinear optimal control

The classical bang-bang theorem is usually stated for linear systems (e.g. Control Theory from the Geometric Viewpoint by Agrachev-Sachkov, p. 209). Sussman proved a nice generalization for systems ...
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0answers
65 views

Diffeological spaces and Sikorski differential spaces

Diffeological spaces and Sikorski differential spaces are each a generalisation of a smooth manifold. In their definitions, both have locality and smooth compatibility conditions. Any diffeological ...
5
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1answer
139 views

harmonic coordinates on non-compact manifolds

Is it possible to show the existence of harmonic coordinates (e.g., on uniform-sized balls) on certain classes of non-compact Riemannian manifolds? For example, one may expect that such harmonic ...
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1answer
95 views

Holonomy of a Warped Product Metric

A warped product metric on the "cone" $\tilde{M} = \mathbb{R}^{+} \times M$ is $\tilde{g} =dr^2 + r^2g_M$ where $g_M$ is the metric on $M$. If we know the holonomy group of the manifold $(M,g_M)$, ...
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Subspace of smooth Foliations inside smooth Distributions - Is it a Frechet submanifold or atleast a Retraction?

Assume M is a closed manifold. Does the set of smooth foliations on M form a Frechet submanifold inside the Frechet manifold consisting of smooth distributions? If not, can the set of smooth ...
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38 views

Metrically homogeneous spaces as inverse limits

Let $(X,d)$ be a locally compact, separable, connected and $\sigma$-compact metric space such that the group of isometries $G$ acts transitively on $X$. The question is the following: Is $X$ ...
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1answer
223 views

Bieberbach theorem for compact, flat Riemannian orbifolds

In his thesis, Bieberbach solved Hilbert 18 problem and proved that any compact, flat Riemannian manifold is a quotient of a torus. I need a reference to an orbifold version of this result: any ...
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0answers
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foliations of a manifold [duplicate]

Let $M$ be an $n$-dimensional open manifold. We assume that there are two compact sets $K_1$ and $K_2$ of $M$ such that $M\backslash K_1$ is diffeomorphic to $N_1 \times (0,1)$ and $M\backslash K_2$ ...
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69 views

$T\bar{T}$ deformation: Stress-energy momentum tensor deformed in CFT and in QFT for various $d$-dimensions

The $T\bar{T}$ deformation is based on the original work of Zamolochikov [1] explored deformations of two-dimensional conformal field theories (CFT) by an operator that is quadratic in the stress-...
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0answers
56 views

Vector bundle endomorphism diffeomorphism invariant?

Let's say we have a vector bundle $V$ on a compact Riemannian manifold $M$ (of dimension $m$, with metric $g$). Given a differential operator $P$, acting on the sections of $V$, of the form: $$P = \...
5
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2answers
194 views

Convexity in co-ordinate charts of geodesic balls

Let $g_{ij}$ be a Riemannian metric tensor on an open subset $U\subseteq \mathbb{R}^n$, and let $p\in U$. I would guess the following is true: for $\epsilon$ sufficiently small, the $g$-geodesic ...
4
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0answers
46 views

Compact, incomplete semi-Riemannian manifold of constant curvature

In the Riemannian setting, Hopf-Rinow tells us that any compact Riemannian manifold is complete. The Clifton-Pohl torus gives a counter example for indefinite metrics. However, in the Lorentz setting,...
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273 views

Homogeneous Riemann Surfaces

A Riemann surface $X$ is a connected complex manifold of complex dimension one. A homogeneous space is a manifold with a transitive smooth action of a Lie group. I guess there must be a classification ...
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0answers
123 views

Holomorphic version of Darboux's theorem

I would like to ask if there is a holomorphic version of Darboux's theorem. More concretely, given a holomorphic symplectic manifold $(X, \omega)$ is there a local holomorphic symplectomorphism from $(...
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0answers
91 views

Foliated vector bundle and basic connection

Let $(M,F)$ be a Riemannian foliation, i.e. there is a metric of $TM$ such that $g$ is bundle-like (locally $g_Q=g_{ij}(y)\,dy^i\otimes dy^j$ for a foliated chart $(x,y)$ and $Q=TM/F\cong F^\perp$). ...
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1answer
66 views

A question concerning the developing map of (G,X) manifolds

Let $M$ be a $(G,X)$ manifold, that is we have local charts $(U,\varphi_U)$ on $M$ with $\varphi_U$ a diffeomorphism onto an open subset of $X$ and the transition maps are locally-$G$. Let $\mathfrak{...
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0answers
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Dichotomy of Riemannian holonomy groups

Berger's list of irreducible Riemannian holonomies contains two sorts of holonomy groups. The first ($SO(n)$, $U(m)$ and $Sp(k)\cdot Sp(1)$) consists of holonomy groups of rank one symmetric spaces (...
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1answer
85 views

Inducing linear connections via functors

Let $M$ be a smooth manifold and let $\pi:E\rightarrow M$ be a real vector bundle over it. Let $\nabla$ be a linear (Koszul) connection on $E$ (here in this question I am using covariant derivatives, ...
15
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1answer
283 views

Is the subgroup $\mathrm{Diff}(M,S)$ of $\mathrm{Diff}(M)$ a Lie subgroup?

Denote by $\mathrm{Diff}(M)$ the Lie group of smooth diffeomorphisms on a compact smooth manifold. Its Lie algebra can be viewed as the Lie algebra $\mathfrak X(M)$ of vector fields on $M$. Now, given ...
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0answers
77 views

Manifold with no closed components?

Let $M$ be a manifold with boundary. Reading some papers on $3$-manifolds I have come across some statements where they require that: ”$M$ has no closed components.” What does this mean? The ...
2
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1answer
110 views

Poincare constant under Ricci curvature lower bound

Let $\mathbf{M}$ be a submanifold of $\mathbb{R}^n$ with the induced Euclidean metric, and $\mbox{Ricc} \geq - \kappa , \kappa \geq 0$, as well as diameter bounded by $D$. What is the best known ...
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0answers
65 views

non-self-intersecting geodesics

Suppose $(M,g)$ is a smooth compact orientable Riemannian manifold of dimension $d \geq 3$ with a smooth boundary $\partial M$ and let $\gamma$ be a maximal geodesic in $M$ starting from a point $p \...
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1answer
94 views

An asymptotic version of the Isoperimetric inequality

Let $U$ be a simply connected bounded open set in $\mathbb{R}^2$. The area of $U$ is denoted by $A$. (We do not assume any thing about its boundary). Assume that $\gamma_n$,s are smooth simple ...