# Questions tagged [dg.differential-geometry]

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

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### 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

**1**answer

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

**1**answer

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

**0**answers

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

**1**answer

121 views

### Moishezon manifold with vanishing $b_2$

Does there exist a closed Moishezon manifold with zero second Betti number?

**5**

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**0**answers

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

**1**answer

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

**0**answers

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

**0**answers

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

**0**answers

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

**0**answers

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

**2**answers

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

**1**answer

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

**0**answers

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

**0**answers

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

**0**answers

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

**1**answer

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

**0**answers

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

**2**answers

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

**0**answers

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

**1**answer

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

**0**answers

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

**1**answer

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

**0**answers

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 ...

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votes

**0**answers

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

**1**answer

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

**1**answer

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

**1**answer

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

**0**answers

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

**0**answers

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

**2**answers

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

**3**answers

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

**2**answers

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

**2**answers

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

**1**answer

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

**7**answers

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

**0**answers

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

**0**answers

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

**0**answers

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

**0**answers

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

**1**answer

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

**0**answers

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

**0**answers

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

**0**answers

18 views

### Relation of complex line bundles and circle bundle [migrated]

What is the relation of complex line bundles to circle bundles?

**3**

votes

**0**answers

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

**1**answer

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

**1**answer

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

**0**answers

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

**1**answer

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

**0**answers

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 $\mathbb{R}^8$?