Questions tagged [dg.differential-geometry]
Complex, contact, Riemannian, pseudo-Riemannian and Finsler geometry, relativity, gauge theory, global analysis.
8,645
questions
2
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Density of continuous functions to interior in set of all continuous functions
Let $M$ be an $m$-dimensional manifold and $N$ be an $n$-dimensional manifold with boundary. Suppose also that the topology on $N$ can be described by a metric. Thus, the set $C(M,N)$ can be endowed ...
1
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0
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33
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Reference request: Transverse parabolic Schauder estimates
Is there a version of the parabolic Schauder estimates for transversely parabolic linear PDE's on a manifold with a Riemannian foliation for functions that are constant on the leaves of the foliation? ...
8
votes
1
answer
700
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Cobordism invariants: topological v.s. geometric
Some cobordism invariants are not cohomology classes. Such as the $\mathbb{Z}_{16}$-valued eta invariant $\eta$ of $\Omega_4^{Pin^+}$, the $\mathbb{Z}_8$-valued Arf-Brown-Kervaire invariant ABK of $\...
5
votes
2
answers
554
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Geometry of Level sets of elliptic polynomials in two real variables
Updated:
A polynomial $P(x,y)\in \mathbb{R}[x,y]$ is called an elliptic polynomial if its last homogeneous part does not vanish on $\mathbb{R}^2\setminus\{0\}$.The two answers to this post provide a ...
2
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0
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199
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Elementary questions about vanishing cycles and emerging cycles
Let $X\to D$ be a proper $C^\infty$ map with $D$ an open disk about the origin in some Euclidean space. Suppose $0\in D$ is the only singular value, i.e that over $D^\times=D\setminus \left\{ 0 \right\...
1
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0
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115
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Defining the cospecialization in topology
Below is an excerpt from part V of Deligne's Étale cohomology - starting points.
Let $X$ be a complex analytic variety and $f:X\to D$ a morphism from $X$ to the disk. We denote by $[0,t]$ the ...
2
votes
1
answer
648
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Riemann-Hilbert correspondence versus Simpson correspondence
I couple of days ago, I asked extensively the same question on Stack-exchange (see https://math.stackexchange.com/questions/3592151/riemann-hilbert-correspondence-versus-simpson-correspondence)and go ...
6
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2
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553
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The set of embeddings is open in the strong Whitney topology
In Hirsch's book "Differential Topology," he claims in Chapter 2, Theorem 1.4 that the set of $C^1$-embeddings is open in the strong Whitney topology $C^1(M, N)$ where $M$ and $N$ are $C^1$ manifolds. ...
13
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2
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871
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Example of two exotic closed 4-manifolds s.t. SW(X)=0
I am interested in seeing examples of two closed 4-manifolds $X_1,X_2$ such that $SW(X_i)=0$ and they are homeomorphic but not diffeomorphic.
So far in the literature I've only found examples which ...
6
votes
1
answer
427
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Hyperbolic manifolds with infinite cyclic fundamental group
It is a well known fact that there is a correspondence between complete hyperbolic $n$-manifolds up to isometry and discrete subgroups of isometries of the hyperbolic space $\mathbb{H}^n$ that act ...
4
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2
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306
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Riemannian manifolds which admit a smooth free $\mathbb{Z}/3\mathbb{Z}$ action but do not admit an equilateral triangle action
A free action of $\mathbb{Z}/3\mathbb{Z}$ on a Riemannian manifold $(M, g)$ is called an equilateral action if for every $x\in M$ all three points of orbit of $x$ have the same distance from each ...
1
vote
1
answer
107
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Chart in $1$-parameter family of Lagrangians in a Kähler manifold
Let $(X,\omega,J)$ be a complex $n$-dimensional Kähler manifold ($\omega$ Kähler form, $J$ complex structure) and $L \subset X$ be a closed real-analytic Lagrangian submanifold. Furthermore, let $L_{t}...
34
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7
answers
15k
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geometric interpretation of Lie bracket
On page 159 of "A Comprehensive Introduction To Differential Geometry Vol.1" by Spivak has written:
We thus see that the bracket $[X,Y]$ measures, in some sense, the extent to
which the integral ...
6
votes
3
answers
911
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How should one think about the band of a gerbe?
Let $X$ be a topological space. Let $\mathcal{F}$ be a fibered category over $X$; seen as an assignment of a category $\mathcal{F}(U)$ for each open $U\subseteq X$.
A fibered catgeory $\mathcal{F}$...
3
votes
0
answers
390
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Pullback connection and diffeomorphism of the base
Let $p \colon E \to B$ be a vector bundle, $\nabla^E \colon E \to E \otimes \Omega^1_B$ a connection on $E$, and $\phi \colon B \xrightarrow{\sim} B$ a diffeomorphism. Further, let there be a natural (...
8
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0
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497
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A cohomology associated to a (not necessarily integrable) distribution (Hilbert 16th problem and dynamical Lefschetz trace formula 2)
Let $(M,D)$ be a pair consisting of a manifold $M$ and a distribution $D$ on $M$. The de Rham complex $\Omega^*(M)$ has the following subcomplex $$\Omega^*(M,D)=\{ \alpha \in \Omega^*(M)\mid \alpha_{|...
0
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0
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207
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Tangent bundle of symmetric product of surface
Hello everybody please help me with this doubt:
Let $f:\mathbb{P}^{1} \rightarrow \mathrm{Sym}^{d}(X)$, where $X$ is a smooth projective surface, $\mathbb{P}^{1}$ is a projective line and $\mathrm{...
2
votes
1
answer
449
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Katz's paper on $p$-curvature – help with proof understanding
I am studying N. Katz's paper "Nilpotent connections and the monodromy theorem: applications of a result of Turrittin" where I found a fairly good account on $p$-curvatures.
I don't understand the ...
1
vote
1
answer
243
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Injectivity radius with respect to continuous change of metric
Suppose $M$ is a smooth manifold and for each $t\in [0,1]$ let $g_t$ be a Riemannian metric on $M$ such that $t\mapsto g_t$ is continuous. If $(M,g_0)$ has positive injectivity radius, does that imply ...
0
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0
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143
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Making area/volume calculations that use SIA rigorous
There are some intriguing "proofs" using Smooth Infinitesimal Analysis of theorems concerning areas and volumes. Some examples:
A proof that $\sin'(0) = 1$.
A proof that the surface area of a cone is ...
5
votes
0
answers
431
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A struggle with jets and Grothendieck vs Ehresmann connections
Let $X\to Y$ be a $C^\infty$ submersion. Consider the following two sheaves.
The sheaf on $Y$ comprised of jets of sections of $X\to Y$.
The sheaf on $X$ given by the quotient of $\Delta_{X/Y}^{-1}C^\...
12
votes
1
answer
902
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Aren't Riemannian geodesics also geodesics of the associated Cartan geometry?
I was inspired by R. W. Sharpe's book on doing differential geometry through Cartan connections. Unfortunately, the book is fairly thin in terms of specific examples in Riemannian geometry, so I ...
8
votes
0
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330
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Differential version of $G\mapsto H^3(G,\mathbb Z)$?
Let $\mathit{cLieGrp}^{\mathrm{inj}}$ be the category of compact connected Lie groups, and injective continuous group homomorphisms.
Is there a reasonable functor (some kind of degree $3$ differential ...
2
votes
0
answers
56
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Lie groupoid $G$ extensions and principal $\text{Out}(G)$ bundles over Lie groupoids
I am reading the paper Non abelian differentiable gerbes by C. Laurent-Gengoux et.al.
The definition $3.4$ of the paper goes as follows:
Definition : Let $X_1\xrightarrow{\phi} Y_1\...
7
votes
3
answers
443
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Why the third stage of Cech nerve a Lie 2-groupoid?
In the page https://ncatlab.org/nlab/show/Lie+2-groupoid the Lie 2-groupoid is defined as the 2 truncated $\infty$-Lie groupoid.
I am not much comfortable with the language of higher category theory ...
8
votes
2
answers
442
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The action of $GL_{\infty}$ on the infinite wedge space
This is a question from the book "Highest weight representations of infinite dimensional Lie algebras, 2nd ed" by V. G. Kac, A. K. Raina, and N. Rozhkovskaya.
Consider the following objects:
the ...
3
votes
0
answers
139
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$\mathbb{Z}_2$-grading by Hodge star operator (for signature theorem)
This question may be a bit low level for MO but I have not received any attention from the SE post.
Consider the algebra of exterior forms $\bigwedge T^*M$ on an even dimensional $n$-manifold $M$. We ...
5
votes
1
answer
191
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Reeb stability counterexample: foliation in $S^{n-2}\times S^1\times S^1$ with non-diffeomorphic leaves
Reeb's global stability theorem requires the foliation to be of codimension 1. As a counterexample, in "Geometric theory of foliations", Camacho and Lins Neto present the following.
Consider the ...
1
vote
0
answers
114
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connections on Lie groupoids/differentiable stacks
Let $(\Gamma_1\rightrightarrows \Gamma_0)$ be a Lie groupoid.
There are many places which define the notion of connection on a Lie groupoid.
As far as I have seen, there is no mention of these ...
14
votes
1
answer
992
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Egg-ovoid rolling down an inclined plane
I am seeking a mathematical analysis of an egg-ovoid rolling down an inclined plane,
for pedagogical reasons.
It is well-known folk lore that the shape of an egg prevents it from rolling away from
...
2
votes
1
answer
219
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Bounding the determinant of the Jacobian between a set and its polyhedral approximation
My question is, essentially, suppose I have two simply connected subset of $R^n$, if I know that the boundary's of both are very close, how can I bound the determinant of the Jacobian between them. I ...
0
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0
answers
71
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Quasi Riemannian submersion and retraction
Let $M, N$ be Riemannian manifolds. A smooth submersion $f:M \to N$ is called a quasi Riemannian submersion if for every $x\in M$ the restriction of linear map $Df_x$ to orthogonal complement of $\...
5
votes
2
answers
304
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Is this subset of matrices contractible inside the space of non-conformal matrices?
Set $\mathcal{F}:=\{ A \in \text{SL}_2(\mathbb{R}) \, | \, Ae_1 \in \operatorname{span}(e_1) \, \, \text{ and } \, \, A \, \text{ is not conformal} \,\}$, and
$\mathcal{NC}:=\{ A \in M_2(\mathbb{R}) \...
2
votes
0
answers
47
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Solving equations of motion of holomorphic BF theory - pure gauge in complex coordinates
In this paper by Bailieu and Tanzini, aspects of holomorphic BF theory are presented.
Holomorphic BF theory on a four dimensional Kahler manifold is discussed from page 5, and on page 8 the ...
2
votes
1
answer
287
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Riemannian submanifolds of Euclidean space admitting Lipschitz extension of Lipschitz functions, and converse statement
Let $M \subset \mathbb{R}^p$ be a Riemannian submanifold. In what follows, when we talk about a Lipschitz function $f$ on $M$, namely $f: M \to \mathbb{R}$, we will assume there is a $L > 0$ so ...
2
votes
0
answers
61
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Articles about Mather's Geometric Groups?
I'm trying to find some information about "Mather's Geometric Groups". But the information on that subject is quite scarce, the only thing I found was the "Mather's Geometric Lemma" in the book "Local ...
-1
votes
1
answer
177
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Reparameterization and group structure
I ran into the following question; let $x,y$ be two points in $\mathbb{R}^d$. Let $(\psi_t)_{t\geq 0}$ be the mapping from $\mathbb{R}^{2d}$ to $\mathbb{R}^{2d}$ defined, for all $t\geq 0$, by
$$
\...
2
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0
answers
58
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Is keeping the kernel fixed an open condition for maps of vector bundles?
More precisely, let $M$ be a smooth manifold, $E_1$, $E_2$ vector bundles over $M$, and consider a $C^\infty(M)$-linear map $A:\Gamma(E_1) \to \Gamma(E_2)$ of vector bundles.
Now consider the ...
1
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0
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61
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Minimal radius of a ball admitting a trivialization of a vector bundle
Let $X$ be a compact Hausdorff space and $p : V \to X$ a complex vector bundle of rank $n$. For $r > 0$ let $B(r,x)$ denote the open ball of radius $r$ around $x$. Does there exist an $r$ such that,...
2
votes
2
answers
1k
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From Riemannian curvature to Ricci curvature in warped product manifold
Let $M=B \times_f F$ be a warped product of two pseudo-Riemannian manifolds. If $X, Y, Z \in L(B)$ and $U, V, W \in L(F)$, (with $L(B)$ and $ L(F)$ I mean the set of all horizontal and vertical lift ...
4
votes
1
answer
1k
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Taylor expansion of determinant of Riemannian metric in normal coordinates up to higher order
Let $(M,g)$ be an $n$-dimensional Riemannian manifold. Let $p\in M$, and let $\{x^i\}_{i=1}^n$ be normal coordinates centered around $p$.
Using Jacobi field, one can show that the metric $g$ has the ...
7
votes
0
answers
196
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Understanding the odd-dimensional index
Given a Dirac operator $D$ on a closed odd-dimensional manifold $M$, I've sometimes heard it said that the Fredholm index of $D$ vanishes because it is an ungraded self-adjoint operator, so that $\dim\...
13
votes
0
answers
705
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Hilbert 16th problem and dynamical Lefschetz trace formula
I would like to apply the known version of the conjectural formula (11) page !0 of the paper Number theory and dynamical Lefschetz trace formula.
Disclaimer: I do not have a complete ...
6
votes
2
answers
430
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How to compute the Kahler potential of a Sasaki metric
The Question
Given Hessian manifold $M$, there is a natural Kahler structures on $TM$. Is it possible to write the Kahler potential of these in terms of the Hessian potential?
Background
To ...
7
votes
0
answers
665
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Nash-Tognoli Theorem
The Nash-Tognoli theorem says that every closed, smooth manifold is diffeomorphic to a real algebraic variety.
Suppose I wanted to study the Ricci curvature of some class of manifolds.
Is there a "...
7
votes
0
answers
244
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Weak 2-groups and non-abelian gerbe over a manifold
In literature, a strict 2-group is defined as a group object in the category of categories or groupoids. It has many well known equivalent descriptions viz:
1. A strict monoidal category in which all ...
10
votes
1
answer
583
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Index of Dirac operator and Chern character of symmetric product twisting bundle
I am having trouble understanding a couple of lines of computation from Theorem 13.30 in Besse's Einstein Manifolds text
We are twisting the spinor bundle (on Einstein 4-manifold) $\Sigma$ with an ...
4
votes
0
answers
321
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Smoothability of open 4-manifolds
F. Quinn proved that any open topological 4-manifold admits a smooth structure in Ends of maps III: dimensions 4 and 5.
He first proves the generalized annulus conjecture:
Suppose $h:D^j\times \...
4
votes
0
answers
226
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Reference request : Quotient manifold theorem for Lie groupoid action on a manifold
Let $G$ be a Lie group and $M$ be a smooth manifold. Let $G\times M\rightarrow M$ be a smooth map giving a free, proper action of $G$ On $M$. Then, by quotient manifold theorem, we see that there ...
2
votes
1
answer
317
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Foliation of tangent bundle arising from exponential map
We first mention our motivation:
For $M=\mathbb{R}$ with usual Riemannian metric the exp map $exp:TM\to M$ is in the form$(x,v)\mapsto x+v$
The level sets of this map define a foliation whose leaves ...