Questions tagged [dg.differential-geometry]
Complex, contact, Riemannian, pseudo-Riemannian and Finsler geometry, relativity, gauge theory, global analysis.
8,629
questions
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Handling Degenerate Planes in Pseudo-Riemannian Geometry: Impact on Sectional Curvature and Comparison Theorems
I've been studying Riemannian and pseudo-Riemannian manifolds and came across an intriguing point regarding the definition of sectional curvature in both geometries.
In pseudo-Riemannian geometry, for ...
2
votes
1
answer
109
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Relationship between Frobenius theorem, curvature, and integrability
In this answer to References for "modern" proof of Newlander-Nirenberg Theorem John Hubbard alluded to something called the "Frobenius integrability form" $\phi\mapsto\bar\partial\...
1
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68
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Extend a circle action on $3$-manifolds
Let $M$ be an oriented closed $3$-manifold equipped with an effective smooth circle action.
Can we have a classification of all such $M$ such that there exists a $4$-manifold $N$ with $\partial N=M$, ...
4
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83
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Regularity of exponential map for $C^{2,\alpha}$ Riemannian metrics
Let $g$ be a $C^{2,\alpha}$ Riemannian metric and $0<\alpha<1$. Would the exponential map $\mathrm{exp}_p$ be $C^{1,\alpha}$ as the point $p$ varies?
Since $\mathrm{exp}_p$ is defined by the ...
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32
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Construct compact submanifold containing non-compact Nash embedded submanifold
$$
\newcommand{\R}{\mathbb{R}}
\newcommand{\geu}{g_{\text{Eu}}}
\newcommand{\X}{\mathcal{X}}
\newcommand{\iX}{\mathring{\X}}$$
Let $\X$ be a closed bounded convex set in some Euclidean space. Its ...
13
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0
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217
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Which G-structures lead to analyticity?
It’s well known that integrability conditions on $G$-structures sometimes force a unique analytic structure on the underlying manifold:
A Lie group structure (a 1-structure with equivariantly ...
2
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0
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38
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Under what conditions principal directions define an integrable distribution?
Consider a hypersurface $M^n \subset \mathbb{R}^{n+1}$ which is compact without boundary. Assume that its second fundamental form $A$ has distinct eigenvalues $\lambda_1<\ldots<\lambda_k$ (with $...
5
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48
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The universal cover and embedding
Whether there exists a non-simply connected complete Riemannian manifold $(M^n, g)$ such that it can be $C^2$-isometric (globally) embedded into some Euclidean space $\mathbb{E}^p$, but its ...
1
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0
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64
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Metric of negative curvature on connected sum
Let $(M_1,g_1)$ and $(M_2,g_2)$ be two Riemannian manifolds of dimension $n\geq 2$. If we consider the connected sum $M=M_1\mathbin{\#}M_2$ of the two manifolds; can one get a smooth metric $g$ on $M$ ...
3
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1
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179
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A complex version of the Cahiers topos
Has anyone tried defining a complex version of the Cahiers topos?
If we take the definition of $C^\infty$-rings, replace "smooth" with "holomorphic" (of course, one has to take ...
2
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189
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+50
Do the eigenvalues of a thin strip around a semicircle converge to the respective eigenvalues of a semicircle?
Background:
Consider the sphere $S^2$ with the round metric $g$ and let $\gamma$ be one half of a great circle of length $\pi.$ Let $T_\epsilon$ denote a geodesically convex tube around $\gamma$ of ...
2
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0
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33
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Growth/Decay of conformal Killing fields in cone metrics
Let $\gamma$ be a smooth metric on $S^2$ of positive curvature. Consider the metric
$$g= dr^2 + r^2 \gamma$$
on $[1,\infty) \times S^2$.
Does there exist a nontrivial conformal Killing field vanishing ...
2
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1
answer
136
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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\...
2
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1
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85
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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 ...
-1
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77
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Zero of gradient of a sum of squares of polynomial
Let $f_1,\ldots,f_n\in \mathbb{R}[x_1,\ldots,x_n]$ have non zero constant jacobian determinant.
Then is it true that $f_1^2+\cdots+f_n^2$ has any critical point?
Let me explain a motivation.
If ...
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138
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Kähler manifold with negative sectional curvature
Goldberg's theorem states that every almost Kähler manifold of constant curvature is Kähler if and only if the curvature is zero. This seems to contradict the fact that the sectional curvature of the ...
1
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0
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68
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Total curvature of a conjugate minimal surface
Let $s: S \to \Bbb R^3$ be an immersed minimal surface with finite total curvature and a proper annular end (possibly with other types of ends).
What is exactly meant by a proper annular end? It is an ...
3
votes
1
answer
240
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Ricci flow and curvature
I am trying to read about geometric flows mainly Ricci flows. I have a question in mind, which I am not sure whether it's possible or not.
So my question is if one starts with a metric that has mostly ...
2
votes
1
answer
166
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Frobenius theorem and the size of integral manifold
Let $X =(X_0,X_1)\in \mathbb{R}^2$ and $Y=(Y_0,Y_1)\in \mathbb{R}^2$ be two vector fields on $\mathbb{R}^2$ such that $X,Y$ are independent on each tangent plane and
$[X,Y]:=XY-YX=0$.
Then by ...
2
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1
answer
81
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Parameterizing Teichmüller spaces of punctured surfaces
Let $S_{g,n}$ denote a genus $g$ surface with $n$ punctures. There is a map $F$ from the Teichmüller space of the punctured surface $T(S_{g,n})$ to the Teichmüller space of the compact surface $T(S_{g}...
9
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303
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Mappings of the sphere (to itself) defined by homogeneous polynomials
Preamble
$\DeclareMathOperator\SO{SO}$Let $\mathbb{S}^m\subset \mathbb{R}^{m+1}$ be the standard unit sphere. An observation of Do Carmo and Wallach states that
If $G$ is a subgroup of $\SO(m+1)$ ...
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0
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81
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Describing the hyperbolic structure of punctured torus in terms of the period lattice
Let $T$ be a torus, $T^* = T - \{p\}$ be the complement of a point $p$. Let's fix a pair of generators $x,y\in\pi_1(T^*)$. Their images in $\pi_1(T)$ also generate, and will also be denoted by $x,y$.
...
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0
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62
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Extending $G$-equivariant local diffeomorphisms on principal bundles to local bundle maps
Consider a principal $G$-bundle $P$ over the base space $M$ equipped with a connection 1-form $\omega$. Let $\mathcal{U}$ and $\mathcal{V}$ be open subsets of $P$, and suppose $F: \mathcal{U} \to \...
5
votes
1
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221
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Bochner Laplacian in coordinates
Sorry if this is a too basic question, but I didn't find an answer anywhere:
The connection Laplacian, or Bochner Laplacian, is the differential operator acting on $k$-tensor fields $T\in\Gamma^{\...
0
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0
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42
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How do I find an algebraic expression for the function $F(ξ, \bar{ξ})$ from this paper?
I am working on understanding the paper "On $C^2$-smooth Surfaces of Constant Width" by Brendan Guilfoyle and Wilhelm Klingenberg. As part of their definition of equations for a 3D surface, ...
2
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0
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45
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Smoothness of the Fréchet Function
Let $M$ be a compact Riemannian manifold and $d$ be the induced distance function. Suppose $\mu$ is a probability measure on $M$ with continuous density. The Fréchet function is defined as
$$
F(x) = \...
7
votes
3
answers
598
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A continuous version of Carathéodory's convex hull theorem
A well-known theorem of Caratheodory states that any point in the convex hull of a set $X\subset R^n$ lies in the convex hull of at most $n+1$ points of $X$. I am wondering about a version of this ...
3
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0
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91
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On the linearized evolution equations in general relativity
The following puzzles me already for quite some time: In mathematical relativity, especially in the discussion of the Cauchy problem, one usually works in the so-called ADM-Formalism, in which one ...
5
votes
1
answer
121
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Regularity requirements for Sard's Theorem
The most common formulation of Sard's Theorem is that for $f\in C^{n-m+1}(\mathbb R^n, \mathbb R^m)$ with $n\ge m$, the set $f(C_f)$ has Lebesgue measure 0, where $C_f=\{x, df(x)=0\}$.
Question. Is it ...
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0
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20
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Is it possible to find the intersection of this involute and roulette, given their parametric equations?
Background
I have two parametric curves, and I want to find the parameter values of their intersection point closest to zero under certain conditions.
The first curve is an involute of a circle with ...
4
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287
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Milnor’s smoothed corners technique for a product of manifolds with boundary and boundary defining functions
Let $M$ be a smooth manifold that we view as the interior of a compact manifold with boundary $\overline{M}$. Let $\rho$ be a boundary defining function for $\overline{M}$, i.e. $\rho$ is smooth, $\...
4
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0
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383
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A 4th-order linear PDE
I am interested in the following type of $4$-th order linear PDE with $2$ variables (i.e., $x$ and $t$):
$x^3 f_{xxxt}+ f =0$
Does anyone know if this type of PDE already appeared in the literature? ...
1
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0
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60
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Does the local family index theorem hold for compact manifolds with corners?
Let $\pi:X\to B$ be a submersion with closed, oriented and spin fibers of even dimension. Suppose $X$ and $B$ are compact, and let $E\to X$ be a complex vector bundle over with a Hermitian metric $g^E$...
2
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0
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89
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+50
soft question, proof sketch: Constructing spaces of analytic sections. silva spaces
This is a follow up on this question Representing geodesic compactifications of $S^1\times \Bbb R$ as analytic sections over base (analytic) foliations to add more details.
In the linked question I ...
2
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1
answer
83
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Coupling small and large injectivity radii
I'd like to know whether a manifold of constant curvature, which has large injectivity radius at many points, can have points of arbitrary small injectivity radius.
More precisely, for a point $x$ in ...
5
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0
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130
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Riemannian structure on connected Hilbert manifolds
The infinite-dimensional separable Hilbert space $H$ has the unusual property that it is diffeomorphic to its unit sphere $S^{\infty}$. Therefore, $H$ admits the round metric as a complete and bounded ...
1
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0
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66
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Representing geodesic compactifications of $S^1\times \Bbb R$ as analytic sections over base (analytic) foliations
Given a smooth nested set of "partial" foliations $\mathcal F_{\alpha}=\big\lbrace e^{\frac{\alpha}{\log x}}: \alpha \in (1/k,k), x\in(0,1),k\in [1,\infty) \big\rbrace$ of $X^2=(0,1)^2$ with ...
0
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0
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72
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Hyperbolicity in sense of Koszul
Question :
I have a question about the notion of hyperbolicity in the sense of Koszul.
Koszul asserts that locally flat compact hyperbolic manifolds do not contain a parallel vector field.
Do you have ...
4
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1
answer
133
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Can every surface be realized as a mean convex hypersurface in $\mathbb{R}^3$?
I'm wondering if every closed surface can be realized as a mean convex hypersurface in $\mathbb{R}^3$, i.e. the mean curvature vanishes or points inward.
Categorizing by genus: for $S^2$ ($g = 0$) ...
1
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0
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44
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Elliptic surfaces with monodromy in Borel subgroup
Are there restrictions on the invariants of an elliptic surface $M\overset{\pi}{\longrightarrow} C$ for the monodromy of its homological invariant to be contained in the upper triangular subgroup of $\...
2
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0
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51
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Is a $C^1$ surface with $C^1$ boundary and uniformly continuous normal a manifold with boundary?
Consider an oriented $C^1$ surface ${\cal S}$ whose closure is a $C^0$ surface with boundary whose boundary is a $C^1$ curve. If the normal to ${\cal S}$ is uniformly continuous, so that it has a ...
2
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0
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157
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When is the closure of a manifold a manifold with boundary?
I'm looking for conditions that ensure the closure of an embedded manifold is a manifold with boundary.
The specific case I'm interested is as follows. Consider an oriented $C^1$ surface ${\cal S}$ in ...
1
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0
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45
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Using connection form for unknown frame field
I have a way to calculate the connection 1-form $\alpha$ associated to a compact simply connected parallelizable Riemannian surface $(M,g)$ (so, $M$ is topologically a disk) and a special orthonormal ...
-4
votes
1
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301
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Does a coarser topology lead to a non-Hausdorff topological manifold? [closed]
Take a topological manifold $M$. Suppose one considers a strictly coarser topology than the manifold topology. Can such topology result in a non-Hausdorff topological manifold?
NOTE: PLEASE avoid the ...
-2
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1
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112
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Is this single-variable function being called with two variables (and if so, how do I handle that), or am I misreading the notation? [closed]
While working on a project, I came across a paper that includes this sum on page 15 as the definition of a support function $r$ for a surface with tetrahedral symmetry:
$$
r(ξ, \bar{ξ}) = \frac{1}{\#𝒢...
7
votes
1
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489
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Conformal Killing fields satisfy a third order PDE
Let $(M,g)$ be a $n$-dimensional Riemannian manifold with smooth metric $g$.
Proposition 3.2 in the paper "The Boost Problem in General Relativity" by O'Murchadha and Chistodoulou claims ...
13
votes
3
answers
2k
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How to get to the earliest time zone?
You are in a plane at some point on Earth. You want to be at the earliest time zone possible at the end of your flight. What is the optimal path to take?
Formally, fix spherical coordinates $(\theta, \...
2
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1
answer
226
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How to chart tubes around manifolds with boundary/corners?
Let $M \subset \mathbb{R}^d$ be a manifold with boundary/corners. For example, a piece of curve with endpoints or a $2d$ unit square in $\{ z = 0 \}$. I am interested in introducing local coordinates ...
2
votes
1
answer
182
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Does the group of compactly supported diffeomorphisms have the homotopy type of a CW complex?
It is known that the group of diffeomorphisms of a compact manifold with the natural $C^{\infty}$ topology has the homotopy type of a countable CW complex. See for instance this thread: Is the space ...
2
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0
answers
372
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Parallel transport on a vector bundle : expansion of the correspondance between normal and tubular coordinates
Let $(M, g^{TM})$ be a Riemannian manifold of dimension $n$. Let $X \subset M$ be a submanifold of dimension $n$ with boundary $\partial X$. Then we have a splitting of the tangent bundle
$$TM \vert_{\...