Questions tagged [dg.differential-geometry]
Complex, contact, Riemannian, pseudo-Riemannian and Finsler geometry, relativity, gauge theory, global analysis.
8,907 questions
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"Modern" proof for the Baker-Campbell-Hausdorff formula
Does someone has a reference to a modern proof of the Baker-Campbell-Hausdorff formula?
All proofs I have ever seen are related only to matrix Lie groups / Lie algebras and
are not at all geometric (...
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How is differential geometry used in immediate industrial applications and what are some sources to learn about it?
Intuitively it might be clear that differential geometry is a very applicable subject in engineering and industry. I'd like to know how some industries/companies use differential geometry. I'd guess ...
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Can one recover a metric from geodesics?
Assume there are two Riemannian metrics on a manifold ( open or closed) with the same set of all geodesics. Are they proportional by a constant? If not in general, what are the affirmative results in ...
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What are the "correct" conventions for defining Clifford algebras?
I have three related questions about conventions for defining Clifford algebras.
1) Let $(V, q)$ be a quadratic vector space. Should the Clifford algebra $\text{Cliff}(V, q)$ have defining ...
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Which almost complex manifolds admit a complex structure?
I was reading Yau's list of problems in geometry, and one of them is to prove that any almost complex manifold of complex dimension $n \geq 3$ admits a complex structure. It's been some time since Yau'...
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Topology of function spaces?
Let $X,Y$ be finite-dimensional differentiable manifolds, and let's assume that they are connected. In fact, in applications I would like both $X$ and $Y$ to be riemannian manifolds.
Let $C^\infty(X,...
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Isometry group of a homogeneous space
Background
Let $(M,g)$ be a finite-dimensional riemannian (or more generally pseudoriemannian) manifold. Suppose that I know that a certain Lie group $G$ acts transitively and isometrically on $M$ ...
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How to define a differential form on a fractal?
It is well known how to construct a Laplacian on a fractal using the Dirichlet forms (see e.g.
the survey article by Strichartz). This implies, in particular, that a fractal can be "heated", i.e. one ...
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What's "geometric algebra"?
Sometimes one bumps into the name "geometric algebra" (henceforth "GA"), in the sense of this Wikipedia article. Other names appear in that context such as "vector manifold", "pseudoscalar", and "...
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Interesting applications of the classical Stokes theorem?
When students learn multivariable calculus they're typically barraged with a collection of examples of the type "given surface X with boundary curve Y, evaluate the line integral of a vector field Y ...
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Manifold embedded in $R^{n+1}$ with a submanifold that doesn't embed in $R^n$
Suppose you have a closed $m$-dimensional manifold $M$, which embeds in $\mathbb{R}^{n+1}$ for some $n$. Can it have a closed submanifold $N$ (of dimension strictly smaller than $m$) which does not ...
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A question about subspace in ${\bigwedge}^2({\mathbb R}^n)$
Let $E$ be a linear subspace of ${\bigwedge}^2({\mathbb R}^n)$. What is the minimal dimension of $E$ that guarantees $E$ contains a nonzero element of the form $X\wedge Y$, with $X, Y\in{\mathbb R}^n$?...
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Largest hyperbolic disk embeddable in Euclidean 3-space?
Hilbert proved that there's no complete regular ($C^k$ for sufficiently large $k$) isometric embedding of the hyperbolic plane into $\mathbb{R}^3$. On the other hand, the pseudosphere is locally ...
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Complex projective manifolds are homeomorphic if homotopy equivalent
If two complex projective manifolds are homotopy equivalent are they homeomorphic?
user164740
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Converse to Stokes' Theorem
Does satisfying Stokes' Theorem imply that a form is linear?
Let $M$ be an $n$-manifold. A differential $k$-form $\omega \in \Omega^k M$ assigns to each point $x \in M$ a function $\omega_x : \Lambda^...
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Example of a compact Kähler manifold with non-finitely generated canonical ring?
A celebrated recent theorem of Birkar-Cascini-Hacon-McKernan and Siu says that the canonical ring $R(X)=\oplus_{m\geq 0}H^0(X,mK_X)$ of any smooth algebraic variety $X$ over $\mathbb{C}$ is a finitely ...
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A paper to the question, if the six dimensional sphere is a complex manifold [duplicate]
for a few days a paper was published on arxiv.org with the title "The six dimensional sphere is a complex manifold": http://arxiv.org/PS_cache/math/pdf/0505/0505634v3.pdf
Because I am not able to ...
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Why should I prefer bundles to (surjective) submersions?
I hope this question isn't too open-ended for MO --- it's not my favorite type of question, but I do think there could be a good answer. I will happily CW the question if commenters want, but I also ...
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Intuition for failure of Implicit Function theorem on Frechet Manifolds
When dealing with moduli spaces of, say connections or metrics, I am using the notions of Frechet spaces/manifolds/groups. I have become familiar with Banach manifolds (I think), but Frechet manifolds ...
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Least number of charts to describe a given manifold
Hello, I'm wondering if there is a standard reference discussing the least number of charts in an atlas of a given manifold required to describe it.
E.g. a circle requires at least two charts, and ...
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Rep Theory Consequences of Bott--Weil--Borel
I've been getting interested in the (Bott--)Borel--Weil theorem lately. As a (mainly) geometer it is very interesting to see representation appearing (from nowhere as far as I can see) in the theory ...
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The "ds" which appears in an integral with respect to arclength is not a 1-form. What is it?
The only reasonable way to interpret "$ds$" as a functional on tangent vectors has to be that it takes a tangent vector and spits out its length, but this is not linear. So $ds$ is not a 1-form. It ...
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What is meant by smooth orbifold?
There seems to be some confusion over what the tangent space to a singular point of an orbifold is.
On the one hand there is the obvious notion that smooth structures on orbifolds lift to smooth $G$-...
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If a triangle can be displaced without distortion, must the surface have constant curvature?
Suppose $S$ is a Riemannian 2-manifold (e.g. a surface in $\mathbb{R}^3$).
Let $T$ be a geodesic triangle on $S$: a triangle whose edges are geodesics.
If $T$ can be moved around arbitrarily on $S$ ...
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Compact surfaces of negative curvature
John Hubbard recently told me that he has been asking people if there are compact surfaces of negative curvature in $\mathbb{R}^4$ without getting any definite answers. I had assumed it was possible, ...
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What's the supersymmetric analogue of the Monster group?
Bosonic string theory lives in 26 dimensions, and it gives a conformal field theory where the field is a map from a Riemann surface to $\mathbb{R}^{24}$. The Leech lattice $L$ is an even unimodular ...
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What results about the topology of manifolds depend on the dimension mod 3?
There are a lot of interesting results about the topology of manifolds that depend on the dimension of the manifold mod 2, mod 4, or mod 8. The simplest ones involve the cup product
$$ \smile \colon ...
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de Rham vs Dolbeault Cohomology
For a complex manifold $M$, one can consider (A) its de Rham cohomology, or (B) its Dolbeault cohomology. I'm looking for some motivation as to why one would bother introducing Dolbeault cohomology. ...
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Penrose’s singularity theorem
Roger Penrose won today the Nobel Prize in Physics for the singularity theorem, which at first glance seems to be a result in pure mathematics.
Questions about the theorem:
What kind of mathematical ...
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Polynomial $g:\mathbb R^n \rightarrow\mathbb R^n$ with no critical point may have no root
Version 1 (solved): If $g$ : $\mathbb R^n \rightarrow \mathbb R^n$ is a polynomial, $Dg(x)$ is non-degenerate for every $x$, then there exists $x$, such that $g(x)=0$.
Version 2: If $f$ : $\mathbb R^n ...
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What is the analogue of simple prime closed geodesic for prime numbers?
The prime geodesic theorem (of Margulis?) states that on a compact surface of (constant?) negative curvature, the number of prime closed geodesics of length at most $L = \log x$ is approximately $e^L/...
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On determinants of Laplacians on Riemann surfaces
History of the formula: In their famous paper "On determinants of Laplacians on Riemann surfaces" (1986), D'Hoker and Phong computed the determinant of the Laplacian $\Delta_n^+$ on the ...
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"The complex version of Nash's theorem is not true"
The title quote is from p.221 of the 2010 book,
The Shape of Inner Space: String Theory and the Geometry of the Universe's Hidden Dimensions
by Shing-Tung Yau and Steve Nadis. "Nash's theorem" here ...
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Important results that use infinite-dimensional manifolds?
Are Banach manifolds (or other types of infinite-dimensional manifolds) just curiosities, or have they been utilized to prove some interesting/important results? Where do they turn up? Important ...
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Topology of SU(3)
$U(1)$ is diffeomorphic to $S^1$ and $SU(2)$ is to $S^3$, but apparently it is not true that $SU(3)$ is diffeomorphic to $S^8$ (more bellow). Since $SU(3)$ appears in the standard model I would like ...
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Most manifolds are hyperbolic?
I heard the claim as in the title for a long time, but can not find the precise reference for this claim, what's the reference with proof for this claim? Thanks for the help.
To be more precise, is ...
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Why is the Leibniz rule a definition for derivations?
In differential geometry, the tangent space is defined as a generalization of directional derivatives, which in turn are defined as functionals following Leibniz's product rule.
I understand all the ...
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Why does the group act on the right on the principal bundle?
In many textbooks, in fact all textbooks I've seen, the fiberwise group action on the principal bundle is on the right. It seems to me that left actions and right actions are essentially the same. ...
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Is there an explicit formula for the hessian of "Determinant"?
Let $f: G= \mbox{GL}(n,\mathbb{R}) \to \mathbb{R}$ be the determinant function.
Then $\mbox{Hess} (f)$ is a two linear map on $M_{n}(\mathbb{R})\simeq T_{e}(G)$ where $e$ is the neutral element of $G$,...
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A simple proof that parallelizable oriented closed manifolds are oriented boundaries?
So let $M$ be a smooth closed orientable real manifold such that $M$ is parallelizable, i.e., the tangent space $TM$ of $M$ is trivial. From the triviality of $TM$ we get that the Stiefel-Whitney and ...
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Conceptual proof of classification of surfaces?
Every compact surface is diffeomorphic to $S^2$, $\underbrace{T^2\#\ldots \#T^2}_n$, or $\underbrace{RP^2\#\ldots \#RP^2}_n$ for some $n\ge 1$.
Is there a conceptual proof of this classification ...
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Rolling without slipping interpretation of torsion
This is, in a sense, a follow up to this question.
Hehl and Obukhov try to give an intuitive description of torsion. I am confused about their description. I am looking at the following paragraph on ...
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Contractibility of the space of Jordan curves
Is the space of Jordan curves in $\textbf{R}^2$ contractible? In other words, is there a canonical or continuous way to deform each Jordan curve to the unit circle $\textbf{S}^1$.
If the curves are ...
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Is the moduli space of unorientable Riemann surfaces with $pin^+$ structure orientable?
By a non-orientable Riemann surface ${\cal C}$, I mean a compact non-orientable two-manifold without boundary that is endowed with a conformal structure.
Such objects have a moduli space that is ...
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Is every closed curve in 3D a geodesic on a genus-0 surface?
Let $\gamma$ be a smooth, closed, unknotted curve embedded in $\mathbb{R}^3$.
Q. Does there always exist a smooth, embedded, genus-zero surface
$S \subset \mathbb{R}^3$
such that $\gamma$ is a (...
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How and why did mathematicians develop spin-manifolds in differential geometry?
First of all, I am neither a physicist nor a mathematician. And I am afraid that mathoverflow is not a suitable place for my question, but having asked similar questions on math SE it is obvious that ...
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Embeddings of $S^2$ in $\mathbb{CP}^2$
Suppose we are given an embedding of $S^2$ in $\mathbb{CP}^2$ with self-intersection 1. Is there a diffeomorphism of $\mathbb{CP}^2$ which takes the given sphere to a complex line?
Note: I suspect ...
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Smooth bijection between non-diffeomorphic smooth manifolds?
The "textbook" example of a smooth bijection between smooth manifolds that is not a diffeomorphism is the map $\mathbb{R} \rightarrow \mathbb{R}$ sending $x \mapsto x^3$. However, in this example, ...
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Does the Gauss-Bonnet theorem apply to non-orientable surfaces?
I hesitated for a long time to ask such an elementary-seeming question on Math Overflow, but when I asked and bountied it on Math SE, I found that a few experts seem to disagree on the answer, and I ...
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Nontrivial tangent bundle that is diffeomorphic to the trivial bundle
Is there an example of a smooth $n$-manifold $M$ whose tangent bundle is nontrivial as a bundle but is nonetheless (abstractly) diffeomorphic to the trivial bundle $M \times \mathbb{R}^n$?
(This ...
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