Questions tagged [dg.differential-geometry]
Complex, contact, Riemannian, pseudo-Riemannian and Finsler geometry, relativity, gauge theory, global analysis.
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4 views
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 ...
2
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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
votes
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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0answers
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.
2
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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 ...
4
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0answers
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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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.
2
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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 ...
2
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0answers
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\...
8
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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{\...
5
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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 ...
2
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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 ...
1
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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 ...
2
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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}(...
7
votes
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/...
1
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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 ...
1
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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 ...
2
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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
votes
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
votes
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 ...
0
votes
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 ...
5
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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 ...
3
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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 ...
2
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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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0answers
27 views
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 ...
2
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0answers
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$ ...
7
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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 ...
1
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0answers
32 views
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$ ...
4
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0answers
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-...
2
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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
votes
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,...
4
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0answers
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 ...
3
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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 $(...
3
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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 $Q=TM/F\cong F^\perp$).
...
2
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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 $\mathfrak{...
4
votes
0answers
79 views
Dichotomy of Riemannian holonomy groups
Berger's list of irreducible Riemannian holonomies contains two sorts of holonomy groups. The first ($SO(n)$, $U(m)$ and $Sp(k)\cdot Sp(1)$) consists of holonomy groups of rank one symmetric spaces (...
4
votes
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
votes
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
votes
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 ...
3
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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 \...
1
vote
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 ...
