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 $\mathbb{R}^8$?
