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Questions tagged [dg.differential-geometry]

Complex, contact, Riemannian, pseudo-Riemannian and Finsler geometry, relativity, gauge theory, global analysis.

7
votes
0answers
88 views

The $\frak{sl}_2$-representation on a symplectic manifold

Any symplectic manifold $(M,\omega)$ carries a representation of $\frak{sl}_2$: Define the maps $$ L: \Omega^\bullet \to \Omega^{\bullet}, ~~~~~~~~~ \Lambda: \Omega^\bullet \to \Omega^{\bullet}, ~~~~~~...
8
votes
1answer
146 views

Moishezon manifold vs proper complex variety

Does there exist a closed Moishezon manifold that does not have the homotopy type of the analytification of a smooth proper complex variety (I think we know that every closed Moishezon manifold is ...
1
vote
1answer
61 views

Existence of meromorphic 2-forms over normal surface singularities

Let $(X,o)$ be an isolated normal surface singularity. Denote by $U:=X\backslash \{o\}$. I am looking for conditions on $(X,o)$ under which there exists a holomorphic section $\omega \in H^0(U, \Omega^...
2
votes
0answers
89 views

Topology of abstract varieties over $\mathbb{C}$

What are the known restrictions on the topology of complex manifolds corresponding to analytifications of smooth proper algebraic varieties over $\mathbb{C}$? I think they have to have non-zero $b_2$ ...
5
votes
1answer
121 views

Moishezon manifold with vanishing $b_2$

Does there exist a closed Moishezon manifold with zero second Betti number?
5
votes
0answers
97 views

Level Sets of Harmonic Maps

Can anybody point me in the direction of some references in which the level sets of harmonic maps between Riemannian manifolds are studied. (Sorry I am unfamiliar with the area and would like some ...
4
votes
1answer
90 views

Is a symmetric, parallel (0,2)-tensor a metric?

I'm interested in affinely connected spaces, on which a metric is not necessarily defined, i.e. $(\mathcal{M},\Gamma)$. Since (as a physicist) my goal is to consider a generalized model of gravity, I ...
2
votes
0answers
30 views

Geometrical regularity of the projection/normalization of a curve

Let $v:\mathbb{R} \rightarrow \mathbb{R}^3/{(0,0,0)} $ be a $C^\infty$ regular arc-length parametrization of a space curve. W.l.o.g. let us assume $v(0)=(1,0,0)$, $v'(0)=(1,0,0)$. Let $\bar{v}$ be ...
3
votes
0answers
58 views

Reference of generalized isometries

I'm wondering if these objects have a name or are studied. No one around me knows, so I thought to ask here. Let $\Phi:\mathbb{R}^d\rightarrow \mathbb{R}^d$ be $C^2$-diffeomoprhism, and fix $p \geq ...
2
votes
0answers
42 views

First eigenvalue of the spherical cap

Let $S$ be the round $n$-sphere of radius $R$ in Euclidean space, and let $r$ be the intrinsic distance from the north pole. Further, let $U(r)$ be the spherical cap of intrinsic radius r. (So $U(0)$ ...
3
votes
0answers
97 views

Explicit KE metrics

Does there exist an explicit example of a Ricci-flat, non-flat metric on a closed manifold? Kaehler--Einstein, non-flat metric on a closed manifold (excluding metrics on homogeneous spaces and ...
4
votes
2answers
163 views

unit element under map of stacks $B\mathcal{G}\rightarrow B\mathcal{H}$

Let $\mathcal{G}$ be a Lie groupoid. The target map $t:\mathcal{G}_1\rightarrow \mathcal{G}_0$ is a principal $\mathcal{G}$ bundle. This article Orbifolds as Stacks? by Eugene Lerman calls (in page $...
3
votes
1answer
83 views

Interpretation of the Schouten bracket as an integrability condition

The Schouten bracket is an extension of the Lie bracket to multivector fields. Given a multivector field $\Lambda$ the vanishing of the Schouten bracket $[\Lambda,\Lambda]=0$ is referred to as a sort ...
12
votes
0answers
247 views

Covering S2 with great circles

Let $S_2$ be the unit 2-dimensional sphere. Is there a way to cover it with great circles such that each point on $S_2$ has 1 or 2 great circles that go through it?
1
vote
0answers
118 views

Interior gradient estimate of mean curvature equation

I'm studying by myself Mean Curvature Flow and I'm reading the paper "Interior estimates for hypersurfaces moving by mean curvature" by Klaus Ecker and Gerhard Huisken. I'm stuck in the theorem $2.3$. ...
-1
votes
0answers
42 views

Submanifold of codimension 1 orientable iff there exists unit normal vector field [closed]

Suppose I have a submanifold $M \subset \mathbb{R}^{n}$, of dimension $n-1$. Where a unit normal vector field is a section $\nu$ of the normal bundle $ TM^{\bot} \to M$. So the fibers are all the ...
7
votes
1answer
238 views

Simple application of Bochner--Weitzenböck type formulas

I am looking for a cool and simple (but not worn-out) application of Bochner--Weitzenböck type formulas which could be used as a motivation. What is your favourite example? P.S. Here is one which ...
2
votes
0answers
124 views

Existence of a certain kind of compact spin manifold with boundary

For a compact spin Riemannian manifold $(M^n,g)$ without boundary, $n \not\equiv 3\mod 4$, it is well-known that the Dirac operator associated with a fixed spin structure $S\rightarrow M$ has real, ...
2
votes
2answers
242 views

Symplectic form on a Kähler manifold can be not real analytic?

Let $M$ be a Kähler manifold. The complex structure on it naturally gives rise to the real analytic structure. I wonder if there exist Kähler manifolds such that the associated symplectic $2$-form $\...
2
votes
0answers
69 views

How to formally connect the log-Euclidean and Riemannian metrics for Symmetric Positive Definite matrices?

I'm a statistician working on a research project dealing with metrics on SPD matrices, specifically the log-Euclidean $d_{LE}(X, Y) = \|\log(X) - \log(Y)\|$ and the Riemannian metric $d_{R}(X, Y) = \|\...
6
votes
1answer
199 views

What is the geometric meaning of one Riemannian metric bigger than the other one on a smooth manifold?

Gromov conjectured in 1985 and LLarull proved in 1998 that: If $g > g_0$ on the sphere, then there exists some point p on the sphere with $Sc(p) < Sc_0(p)$. Here $g, g_0$ are Riemannian metrics ...
19
votes
0answers
216 views

The de Rham complex of the octonionic projective spaces

The complex projective space $\mathbb{CP}^n$ is a complex manifold, and hence its de Rham complex carries a representation of the complex numbers in the form of its complex structure. The quaternionic ...
2
votes
1answer
108 views

Relation between Optimal Transport Cost and Difference between Topological Invariants?

I was working on some mathematics of Wasserstein GAN and found out a seemingly interesting research problem but I am not quite sure whether it has already been studied in some recent literature of ...
2
votes
0answers
137 views

Diagonal is representable then composition is representable

Let $\mathcal{X}$ be a stack over $S$ i.e., a stack over category of schemes over $S$ (which we denote by $Sch/S$) which comes with a functor $\mathcal{X}\rightarrow Sch/S$. Consider the diagonal map ...
0
votes
0answers
78 views

Vector-valued forms in Riemannian geometry

Suppose $(M,g)$ is a Riemannian manifold. I want to find a vector-valued $2-$form $T$ such that, for any vector fields $X,Y,Z$ on $M$, $$ g(T(X,Y),Z)=g(T(Z,X),Y)\,. $$ As a motivation, consider the ...
3
votes
1answer
233 views

Diagonal is representable then any morphism is representable

Ariyan Javanpeykar said here in comments that, If the diagonal is representable, then isn't any morphism $S\rightarrow \mathcal{X}$ with $S$ a scheme representable? I could not find the statement (...
4
votes
1answer
201 views

Existence of solution for the PDE $p \frac{\partial f(p, q)}{\partial p}-q \frac{\partial f(p, q)}{\partial q}=g(p, q)$

Consider the following PDE: \begin{equation} p \frac{\partial f(p, q)}{\partial p}-q \frac{\partial f(p, q)}{\partial q}=g(p, q),\tag{$\star$} \end{equation} where $g$ is a flat function at the point (...
7
votes
1answer
159 views

The developing map of conformally flat manifold

There is one sentence I don't understand in some paper. "A simply connected and conformally flat three mainifold can be conformally immersed into $S^3$" by the means of a developing map. Is any ...
10
votes
0answers
274 views

Fourier transforms and nontrivial vector bundles

We know that in arithmetic, geometry and analysis, Fourier transforms of various forms show up. For example, we have the classical Fourier transform, Fourier-Mukai transforms in the setting of ...
0
votes
0answers
57 views

A Generalized Bernstein's Problem

Suppose $\Sigma\subset \mathbb{R}^n$ be an area-minimizing hypersurfaces, the standard results in GMT tell that when $3\leq n\leq 7$, $\Sigma$ is a hyperplane; while for $n\geq 8$, there are examples ...
5
votes
2answers
92 views

Linearization of hamiltonian torus action

Let $(M,\omega)$ be a symplectic manifold with a hamiltonian effective torus action. Suppose it has an isolated fixed point $p$. Is it true that there exists an invariant neighborhood $U$ of $p$ such ...
7
votes
3answers
317 views

Sectional curvature of leaves of foliation

Given a $k$- dimensional foliation $F$ of a riemannian $n$-manifold $M$, with the property that the leaves of the foliation have constant sectional curvature $s$, for some $s$, is it true that $M$ ...
9
votes
2answers
325 views

Deforming metrics from non-negative to positive Ricci curvature

Given a closed Riemannian manifold $(M,g)$ with non-negative Ricci curvature and $dim\geq 3$, when can we deform the metric to a positive Ricci curved one? I know it's impossible in general due to ...
5
votes
2answers
158 views

Equivalence of two definitions of jets of smooth functions

In the literature I have encountered two different definitions of jets of smooth functions, and I was wondering how one could identify these definitions. One definition is the often encountered ...
14
votes
1answer
643 views

Hadamard theorem about embedding

The following theorem is commonly attributed to Jacques Hadamard. Assume $\Sigma$ is a smooth locally convex immersed surface in the Euclidean space. Then $\Sigma$ is embedded and bounds a convex ...
21
votes
7answers
2k views

Smooth functions on sphere

Let $u$ be a smooth function defined on the unit sphere $S^2$. Assume $u$ has two local maxima, two local minima, and two saddle points (a total of 6 critical points). Does there exist a plane $P$ ...
3
votes
0answers
58 views

Is this pullback of non-degenerate form invertible?

For a fiber bundle $M\longrightarrow N$ where $\dim N=n$, a non-degenerate 1-form $\theta$ on $M$ generates the differential ideal $\mathcal{I}$, and the Lagrangian $\mathcal{L}$ is an $n$-form on $M$....
7
votes
0answers
117 views

Eta-Invariant and Atiyah-Patodi-Singer Index Theorem

In Quantum Field Theory and Jones Polynomial (equation 2.16), Witten used a formula relating the APS eta-invariant to the Chern-Simons action. Witten claimed that it is derived from the Atiyah-Patodi-...
3
votes
0answers
54 views

Maximally symmetric affine manifold

As a physicist who knows (something) about General Relativity, I'm accustomed to the term "maximally symmetric space" being an $n$-dimensional manifold with $\frac{n(n+1)}{2}$ Killing vectors. A ...
1
vote
0answers
78 views

Research trends on mean curvature flow

It's been a few months since I've been curious about mean curvature flow and now I'm reading Robert Haslhofer's lecture notes. I like the subject and would like to do research on it, but I know ...
5
votes
1answer
131 views

Killing vector fields of a conformally flat Riemannian metric

Let $f: \mathbb{R}^n \to \mathbb{R}$ be a smooth function and let's consider the conformally flat Riemannian metric $g = e^f \delta_{ij} dx^idx^j$ on $\mathbb{R}^n$. Is it true that the Killing ...
3
votes
0answers
60 views

Local planar sections of surface are circles then surface is a sphere?

Is the following true: Let, $S$ is a compact connected $(n-1)$-dimensional surface in $\mathbb{R}^n$ s.t., for every point $p \in S$ there is a neighborhood $\mathcal{V}_p \subset \mathbb{R}^n$ of ...
4
votes
0answers
106 views

Trace free Codazzi Tensor on Hyperbolic manifolds

Does there exist trace free nontrivial symmetric Codazzi Tensor on closed manifold with constant sectional curvature -1? I know, locally all Codazzi Tensors on closed manifold $(M, g)$ with constant ...
0
votes
0answers
18 views

Relation of complex line bundles and circle bundle [migrated]

What is the relation of complex line bundles to circle bundles?
3
votes
0answers
91 views

An upper bound for the number of singularities of a transversal vector field isometric to the zero field

Let $(M,g)$ be a Riemannian manifold. We equip the tangent bundle $TM$ with the Sasaki metric $g_s$. A smooth vector field $X:M \to TM$ is called a transversal vector field if $X(M)$ is transverse ...
6
votes
1answer
143 views

Grassmannians of planes isotropic with respect to general tensors

In symplectic geometry, the Grassmannian of isotropic planes for a symplectic vector space is a well known and well studied object; for example, one can realize it as a homogeneous space with a known ...
3
votes
1answer
68 views

Bernstein type theorems for CMC hypersurfaces in $\mathbb{R}^{n+1}$

Is there any Bernstein type theorems for CMC hypersurfaces in $\mathbb{R}^{n+1}$ in the literature? More precisely I would like to know if there is an answer to the following QUESTION: Let $f : \...
4
votes
0answers
96 views

Geometric interpretation for the Lebesgue-Radon-Nikodym Theorem

Discussing with some friends, the following question arose: If $\nu$ is a signed measure, $\mu$ is a positive measure, and they're both $\sigma$-finite, then we may write $\nu = \lambda+\rho$, where $...
6
votes
1answer
115 views

Explaining patterns in modular multiplication graphs

Let the multiplication graph $n/m$ be the graph with $m$ points distributed evenly on a circle and a line between two points $a$, $b$ when $an \equiv b\operatorname{mod} m$. These graphs often look ...
2
votes
0answers
209 views

Embedding of $CP^2/CP^1$ into euclidean space [closed]

Is there a "nice" embedding of $\mathbb{C}\mathbb{P}^2/\,\mathbb{C}\mathbb{P}^1$ into $\mathbb{R}^8$?