# Questions tagged [dg.differential-geometry]

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

**1**

vote

**0**answers

34 views

### A non-vanishing vector field on $S^3$ whose flow does not preserve any transversal foliation

Is there a non-vanishing vector field $X$ on $S^3$ which does not admit a transversal $2$-dimensional foliation? if the answer is negative, is there a non-vanishing vector field $X$ on $S^3$ which ...

**2**

votes

**0**answers

60 views

### Irrational closed orbits of vector fields on $S^2$ (Limit cycles and trace formula)

Motivations: We first introduce our motivations: We wish to find an operator-theoretical interpretation for the number of limit cycles of a polynomial vector field on the plane.
Via Poincaré ...

**2**

votes

**1**answer

62 views

### Projection of an invariant almost complex structure to a non-integrable one

My apologies in advance if my question is obvious or elementary.
We identify elements of $S^3$ with their quaternion representation $x_1 + x_2i + x_3j + x_4k$. We consider two independent vector ...

**-2**

votes

**0**answers

41 views

### Radius of curvature question [on hold]

Im trying to figure out an equation from geometric geodesy. But doing the derivation of that equation results in a different equation then the one from the literature. Its the (1-e^2). I constantly ...

**1**

vote

**0**answers

52 views

### Existence of nonparabolic ends

Let $M$ a nonparabolic Riemannian manifold. If exists only one nonparabolic end $E$. We would like to know why the subspace of space of bounded harmonic functions with finite Dirichlet integral is the ...

**3**

votes

**0**answers

96 views

### Asymptotic bound on minimum epsilon cover of arbitrary manifolds

Let $M \subset \mathbb{R}^d$ be a compact smooth $k$-dimensional manifold embedded in $\mathbb{R}^d$. Let $\mathcal{N}(\varepsilon)$ denote the minimal cardinal of an $\varepsilon$-cover $P$ of $M$; ...

**6**

votes

**0**answers

89 views

### Hermitian sectional curvature

Let $N$ be a Riemannian manifold, denote $R$ its purely covariant Riemann curvature tensor with sign convention so that the sectional curvature is $K(X,Y) = R(X,Y,X,Y)$ for an orthonormal pair.
...

**1**

vote

**0**answers

67 views

### Subdividing a Compact Bounded Curvature Manifold into Charts with Bounded Lipschitz Constant

Let $M \subset \mathbb{R}^d$ be a compact smooth $k$-dimensional manifold embedded in $\mathbb{R}^d$. Let $\mathcal{N}(\epsilon)$ denote the size of the minimum $\epsilon$ cover $P$ of $M$; that is ...

**3**

votes

**1**answer

74 views

### Isomorphism between the hyperbolic space and the manifold of SPD matrices with constant determinant

I'm studying properties of the manifold of symmetric positive-definite (SPD) matrices and I've learnt about the following connection to the hyperbolic space [1, Section 2.2], $$\mathcal{P}(n) = \...

**1**

vote

**0**answers

85 views

### Counting fixed points for Hamiltonian symplectomorphisms on $T^{2}$

This question is motivated by the Lorenz curve used in economic analysis and also the Penrose diagram used in general relativity, used by physicists in order to visualise causal relationships in ...

**3**

votes

**1**answer

88 views

### Reduction of the structure group of $\mathbb{R}^n$-fiber bundles to a special subgroup of $\mathrm{Homeo}(\mathbb{R}^n)$

Let $G$ be the group of all self-homeomorphisms $f$ of $\mathbb{R}^n$ which satisfy $$f(x+m)=f(x)+m,\quad \forall m\in \mathbb{Z}^n.$$
In other words, $G$ is the group of all equivariant self-...

**4**

votes

**0**answers

60 views

### Is there a representation theoretic way to define the pullback of densities and differential forms?

I find it convenient to define the bundle of densities of weight $\alpha$,say $\Omega_\alpha(M)$ over a smooth manifold $M$ as the associated vector bundle of the frame bundle $F(M)$ with the ...

**4**

votes

**0**answers

123 views

### On the definition of the Reeb foliation

To define the Reeb foliation on the sphere, one needs to fix two even functions of the from $f:(-1,1)\to\mathbb{R}$.
In the book I. Tamura, Topology of Foliations: An Introduction, the following is ...

**1**

vote

**0**answers

47 views

### Foliations with algebraic foliation chart

An algebraic foliation chart for a foliated manifold is a foliation chart for which the transition maps are polynomial maps.
What is an example of an analytic foliation of the Euclidean space $\...

**0**

votes

**0**answers

121 views

### Why not hermitian metrics on the tangent bundle?

Let $(M,J)$ be an almost complex manifold. Then $TM$ is naturally a complex vector bundle. Hence it makes sense to consider a hermitian metric $h: TM \otimes \overline{TM} \rightarrow \mathbb{C}$, ...