Questions tagged [dg.differential-geometry]
Complex, contact, Riemannian, pseudo-Riemannian and Finsler geometry, relativity, gauge theory, global analysis.
8,673
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Construction of Scherk's surface using soap films
I am currently interested in the differential geometry of minimal surfaces, and I have a rather trivial question regarding Scherk's surface (the one which can be parametrised by the real function $(x,...
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Existence of a spin map from a standard sphere to any closed Riemaninan manifold with nonnegative curvature operator
Let $S^m$ be a standard sphere of dimension $m=n+4k$, and let $M$ be any closed Riemaninan manifold of dimension $n$ with nonnegative curvature operator.
My question: Is there always a smooth spin map ...
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For proper group action on closed Riemannian manifold, must the union of orbits with non-unique closest points to a given point 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 ...
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Question from Taubes' SW$\Rightarrow$ Gr
I am trying to understand Taubes' paper on SW$\Rightarrow$ Gr. I don't understand how either of the equations 2.16 or 2.17 appears, I would be happy to understand how the curvature term $F_a$ appears ...
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A paper of Borel (in German) on compact homogeneous Kähler manifolds
I am trying to understand the statement of Satz 1 in Über kompakte homogene Kählersche Mannigfaltigkeiten by Borel. Here is the statement in German
Satz I: Jede zusammenhängende kompakte homogene ...
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2
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Are Chern classes always vertical?
Let $c_k \in H^{2k}(M, \mathbb{Z})$ be the $k$-th Chern class of the tangent bundle of a Hermitian manifold $M$.
Is $c_k$ necessarily vertical, i.e.
$$
c_k = \sum_{i_1,\dots, i_{k}} \alpha_{i_1 \dots ...
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A compact Kähler manifold that admits a homogeneous action of a non-reductive Lie group
Is it possible to have a compact Kähler manifold that admits a homogeneous action of a non-reductive Lie group? It seems not to be the case, but a precise argument of reference would be great!
Edit: ...
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Taubes' SW$\Rightarrow$ Gr
I am reading Taubes' paper on SW$\Rightarrow$ Gr and lost in some analysis, can anyone help me to see how to get equation 2.19 from equation 2.18? Is this some version of Kato for the Laplacian?
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A homogeneous manifold that does not admit an equivariant Riemannian metric?
Let $M = G/H$ be a homogeneous space, where $G$ is a Lie group and $H$ is a closed Lie subgroup. Can it happen that $M$ does not admit an invariant Riemannian metric?
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An integration formula that looks like polar coordinates in $\mathbb{R}^n$ [migrated]
Let $M$ be a complete $n$-dimensional Riemannian manifold with non-negative Ricci curvature. Let $x_0\in M$ and $\theta>1$ be fixed. Consider the function $f=\theta^{-1}d(\cdot, x_0)$, where $d$ is ...
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geometry question [closed]
Let M , N , P be points on the sides AB, BC, CA of triangle △ABC.
i)Assume that Q is the second point of intersection of the circumcircles of triangles △BM N
and △N CP . Prove that Q is on the ...
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Differential operators and iterations of tangent bundle
Is there a relationship between higher order differential operators and higher tangent bundle viewed as bundle on the base manifold?
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Germs of left invariant differential operators on a group
Are there germs at the identity of linear differential operators on a group which are not germs at the identity of left invariant differential operators?
I feel like the answer is no but the statement ...
user525442
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Continuity of the volume function
Consider a continuous map $F:(a,b)\times\mathbb{S}^n\to\mathbb{R}^{n+1}$ such that for any $t\in(a,b)$, the map $F(t,\cdot)=F_t:\mathbb{S}^n\to\mathbb{R}^{n+1}$ is Lipschitz continuous. The $n$-...
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Analogue of vector for differential operators
A differential operators of order one is a vector field which is defined pointwise . Differential operators of order greater than one are not. The closest analogue to a vector is given by a germ of a ...
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Index and nullity of a short closed geodesic
Let $g$ be a reasonably smooth Riemannian metric on the n-dimensional sphere $S^n$. Call a closed geodesic $\gamma$ in $(S^n, g)$ short if, for every diffeomorphism $S^n \to S^n$, the image of at ...
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Simply connectedness of leaves of a foliation on an complex manifold
Now I'm searching about leaves of foliation in the following special setting.
Let $U,V$ be two holomorphic vector field on $\mathbb{C}^2$ s.t the Lie bracket $[U,V]=UV-VU=0$ and $U$ and $V$ spaned ...
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Identifying the circle bundle of the canonical line bundle $\mathcal{O}(-n-1)$ over a projective space $\mathbb{CP}^n$
It's not hard to see the following fact: the circle bundle of the tautological line bundle $\mathcal{O}(-1)\rightarrow \mathbb{CP}^n$ is $S^{2n+1}$, the unit sphere inside $C^{n+1}.$ I want to see ...
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Flow of a vector field
Consider a Riemannian manifold $(M^n , g)$ and let $d_p: M^n \to [0,\infty)$ be the distance function of $p \in M^n$. Then the flow lines generated by $\nabla d_p$ are radial geodesics from $p$. Also, ...
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What can we say when a module of differential is free?
Let $\mathbb{C}$ complex number.
$R=\mathbb{C}[x,y]/(f)(f\in \mathbb{C}[x,y])$
If the module of differential $\Omega_{R/\mathbb{C}}$ is free $R$ module of rank one,
what can we say about $R$.
How far ...
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Merits of derived geometry
What are the merits of derived geometry? More precisely, which specific mathematical problems that can be formulated without this machinery have been solved using it? If those problems exist, could ...
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Question about Neumann eigenvalues on manifolds
Question:
Let $\Omega\subset \mathbb{S}^2_+$ be any geodesically convex subset of the hemisphere $\mathbb{S}^2_+$. Then is it true that $\mu_1(\Omega)\geq \mu_1(\mathbb{S}^2_+)$ where $\mu_1$ is the ...
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About geodesic vector fields and the status of a classic problem on the number of closed geodesics
A classical problem in differential geometry is to determine whether every compact Riemannian manifold admits infinitely many geometrically distinct closed geodesics. A good reference on the subject ...
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Converging paths implies converging parallel transports along those paths?
Suppose we have a vector bundle $E$ with connection $\nabla$ over a smooth manifold $M$. Let’s also say we have a sequence of smooth paths $\gamma_n\in C^\infty([0,1],M)$ starting at the same point $\...
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Understanding the integral $\int_0^1\det(v(t),v'(t))dt$ where $v(t)$ is path in the plane
Let $v(t) : [0,1]\rightarrow\mathbb{C}^2$ be a smooth path, and let $v' := dv/dt$. I'd like to understand what the integral:
$$I(v) := \int_0^1 \det(v(t),v'(t))dt$$
tells us about $v$, where $\det(v(t)...
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Condition to show $\{ U \in \mathbb{R}^{n \times p}|\mathscr{A}(UU^{\top}) = b \}$ is (is not) a manifold
Consider $\mathscr{A}: S^{n\times n} \to \mathbb{R}^{m}$, $b \in \mathbb{R}^{m}$, I would like to know when $\mathscr{M}:=\{ U \in \mathbb{R}^{n \times p}|\mathscr{A}(UU^{\top}) = b \}$ is a manifold. ...
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Deriving the definition of vector bundle morphisms from Cartan geometry (a.k.a. why are they linear?)
I'm familiar with the definition of the category of vector bundles, but I'm trying to derive it from some first principles about general fiber bundles. My intuition is that vector bundles should be ...
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What is the image of a smooth map? [migrated]
Let $f: S^2 \to \mathbb{R}^n$ be a smooth map from the two-dimensional sphere to euclidean space. Let $X = \mathrm{Im}(f) \subset \mathbb{R}^n$ be the image topological space (note: the quotient ...
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What are compact manifolds such that GROWTH (of spheres volumes) is well approximated by the Gaussian normal distribution?
Consider some compact Riemannian manifold $M$. Fix some point $p$.
Consider a "sub-sphere of radius $r$" - i.e. set of points on distance $r$ from $p$.
Consider growth function $g(r)$ to be ...
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Using a theorem (which is originally set on 2-dim bounded domain in Euclidean space) on a torus
Actually I'm reading a paper on mean-field equation on torus by M.Struwe and G.Tarantello Here, they studied $$\tag{1}
-\Delta u=\lambda\left(\frac{e^u}{\int_{\Omega} e^u d x}-\frac{1}{|\Omega|}\right)...
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When considering an equation on flat torus, can we treat it as an equation on a square with opposite sites topologically identified? [migrated]
I need to use some theorems (which is originally for $2$-dim bounded domain in Euclidean space) on torus, what should I do to the equation? Can I just simply treat it as an equation on a square with ...
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Naturality of Lie bracket — alternate proof
Let $M$ and $N$ be smooth manifolds, and let $F: M \to N$ be a smooth map. Let $X$ and $Y$ be vector fields on $M$, and let $\tilde{X}$ and $\tilde{Y}$ be vector fields on $N$. We say that $X$ and $\...
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Can a Tangent Space always be expressed with “more structure” than just a vector space (e.g. a choice of basis for Stiefel manifold)
I'm currently trying to read about the Stiefel manifold, or set of all $p$ orthonormal $n$-dimensional vectors embedded in $\mathbb{R}^{n\times p}$.
$$\mathcal{V}_p(\mathbb{R}^n) = \{U \in \mathbb{R}^{...
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Expressing the union of principal orbits as a disjoint union of global slices for proper group actions
Setup:
I was reading about slices and principal orbit theorems (Theorem 3.4.6) from these notes.
Let the Lie group $G$ act on a complete Riemannian manifold $(M,g)$ isometrically on $M$, i.e. $\phi^{*}...
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Approximation of triply periodic minimal surfaces with trigonometric level sets
Some triply periodic minimal surfaces are known to be approximated by trigonometric level sets very accurately.
To see this, let's sample a gyroid scaled to the bounding box $[0, 1]^3$ exactly through ...
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Compactification of a Cartan-Hadamard manifold
Let $X$ be a simply connected manifold with nonpositive sectional curvature. It is standard that $X$ is uniquely geodesic, i.e., for any distinct points $p$ and $q$, there is a unique geodesic ...
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Prove the orthogonality of vector spherical harmonics
We define
$S_a^{lm} = \Big( - \frac{1}{\sin \theta} Y^{lm}_{,\varphi}, \sin \theta\ Y^{lm}_{,\theta} \Big)$
$Y_a^{lm} = \Big( Y^{lm}_{,\theta}, Y^{lm}_{,\varphi} \Big)$
to be the axial vector ...
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Coarse embeddings and Gromov products in (Gromov) hyperbolic spaces
I am new into geometric group theory and I have recently started reading the book "Sur les Groupes Hyperboliques d’après Mikhael Gromov" by Ghys and de la Harpe. The following inequality ...
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Theories of manifolds w/ extra structure and singularities
Many different objects in mathematics can be described as manifolds with extra structure. Among the most famous examples of these are smooth manifolds, Riemannian manifolds, complex manifolds, and ...
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Proving that $H^1(X,\mathcal{Hom}(\mathcal{G},\mathcal{E})) \cong \text{Ext}^1(\mathcal{G},\mathcal{E})$ holds for locally free sheaves
The following passage is from a thesis I'm reading:
Suppose we have a short exact sequence of vector bundles $$0 \to \mathcal{E} \to \mathcal{F}\to \mathcal{G} \to 0.$$ Since these sheaves are ...
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Systems of (hyperbolic) 2nd order PDEs with lower order constraints
Certain surfaces in mechanics are endowed with the fundamental forms
\begin{align}
\text{I} &= \mathrm{d}u^2+\mathrm{d}v^2+2\cos\gamma\: \mathrm{d}u\: \mathrm{d}v \\
\text{II} &= \alpha\left(\...
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Regular value theorem for Banach manifolds without surjectivity
It is well-known (e.g. Lang, Fundamentals of differential geometry, Prop. 2.3 in Chapter II) that the following extension of the regular value theorem holds for Banach manifolds:
Let $\phi : M\...
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Vector bundles over a homotopy-equivalent fibration
I think this question is related to what is known as "obstruction theory", but I'm not very familiar with this field of mathematics, so I am asking here.
Let $\pi:N\rightarrow M$ be a smooth ...
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Convergence of metric implies convergence of eigenvalues?
Consider the metric $g_\varepsilon=d\theta^2 + \varepsilon^2 \sin(\theta)^2 \, d\varphi^2$ on the hemisphere $S^2_+.$ I had two naive questions:
Does $g_\varepsilon$ converge to the flat metric on ...
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Densities, pseudoforms, absolute differential forms and measures, differential forms, etc
Apologies if this question is too basic, but I figured I first heard of most of these concepts on MO, so perhaps I can ask here.
Gelfand’s definition, copied from AlvarezPaiva [My edit, could be ...
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Dolbeault class of the curvature of the Chern connection equaling the Atiyah class
The following is a proposition from Complex Geometry by Huybrechts.
Proposition $4.3.10$. For the curvature $F_\nabla$ of the Chern connection on an hermitian holomorphic vector bundle $(E,h)$ one ...
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Surface terms in the calculus of variations on jet bundles
Let $\pi:N\rightarrow M$ be a fibered manifold with $m=\dim M$ and $m+n=\dim N$. The variational bicomplex on the infinite jet space $J^\infty(\pi)$ is denoted $(\Omega^{k,l}(\pi),\delta,\mathbf d)$ ...
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Does a Riemannian submersion map horizontal geodesics to geodesics, and a relevant question?
I asked this question on MSE, but I didn't receive a response yet, so I'm asking here. Apologies if the question is not exactly a research level question, but I'm having some trouble in figuring them ...
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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 ...
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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\...
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