# Questions tagged [dg.differential-geometry]

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

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### Transitive embedding of the projective space $P^2\Bbb R$ into the $4$-sphere

Is there an embedding (i.e. injective continuous map)
$$\phi:P^2\Bbb R\hookrightarrow S^4\subseteq\Bbb R^5$$
of the 2-dimensional projective space $P^2\Bbb R$ into the $4$-sphere, that is ...

**6**

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**1**answer

60 views

### Integral of top forms in terms of Čech representative

Let $X$ be a compact connected Riemann surface and let $\omega$ be a two-form on $X$. We can view the cohomology class $[\omega]$ as an element of the Čech cohomology group $\check{H}^2(X,\mathbb{R})$,...

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163 views

### A differential operator analogy of certain fact in real analysis of smooth functions

Let $E\to M$ be a smooth vector bundle over a smooth manifold $M$.
Let $D$ be a differential operator defined on the space $\Gamma(E)$ of smooth sections of $E$. We fix a section $s\in \Gamma(E)$.
...

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34 views

### Controlling a Schwartz kernel near the diagonal

Let $D$ be a first-order elliptic differential operator that is essentially self-adjoint on $L^2(\mathbb{R}^n)$. Consider the operator $(D+i)^q$ acting on $L^2(\mathbb{R}^n)$ with domain $C_c^\infty(\...

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35 views

### Calculating the net flux of a field [on hold]

I'm having trouble with the following question:
Consider the vector field F(x)=x/||x||3
Calculate the net flux passing through the plane $z$ = constant
I know the equation to find net flux: $$ \...

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**1**answer

223 views

### Volume comparison on Grassmannian

Let $G(r,n-r)$ be the Grassmannian, which can be identified with the space of all rank $r$ projection matrices in $\mathbb{R}^n$. Let $\mu$ be the uniform measure on $G(r,n-r)$. For any $\lambda_1,......

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55 views

### differential of the riemannian exponential

Let $(M,g)$ be a riemannian manifold with exponential $exp$ and denote its inverse by $log$ and the parallel transport by $\Gamma$. Assuming that $X$ is a curve on $M$, it holds trivially that $exp_{x^...

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31 views

### Numerical methods for geodesics in Lorentz manifolds

There has been much work on the study of geodesic connectedness for Lorentz manifolds (see Analytic approach to geodesic connectedness in Semi-Riemannian manifolds).
Do you know of any references ...

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65 views

### How is the instanton Floer homology of Seifert fibrations related to that of a trivial fibration

My question centers around the relationship of the Chern-Simons theories of a Seifert fibration and the trivial product space $\Sigma_g \times S^1$, and its implication for instanton Floer homology. ...

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193 views

### Circle inscribed between two curves [migrated]

Consider the plane region $S_n$ bounded from above and below for the graphs of $f_n(x)=x^{1/n}$ and $g_n(x)=x^n$, $0\le x\le1$.
How to find the radious and center of the circle inscribed in $S_n$?
...

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83 views

### Volume ratio of balls in Grassmannian with different metric

Let $G(r,n-r)$ be the Grassmannian, which can be identified with the space of all rank $r$ projection matrices in $\mathbb{R}^n$. Let $\mu$ be the uniform measure on $G(r,n-r)$. For any $\lambda_1,......

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**1**answer

118 views

### Generalized Hodge Decomposition on Manifolds with Boundary

This question is motivated by the problem of finding heat kernels to use for the renormalization of quantum field theories on manifolds with boundary.
If $(\mathscr{E}, Q)$ is an elliptic complex on ...

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167 views

### How to prove that the equation is not possible [closed]

I came across another very complex equation (calculating the Gaussian curvature of a surface):
\begin{align*}\frac{-m}{2}=&(2A^2+A)(Du+S-T)^3u^{(3+6A+4B)}\\
&+AD(Du+S-T)^2u^{(6A+4B+4)}\\ &...

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**1**answer

136 views

### Notational question about quadratic differentials in Strebel's book “Quadratic differentials”

In Kurt Strebel's book "Quadratic Differentials", in Chapter 2, $\S4$, he begins by saying:
"Every analytic function $\varphi$ is a domain $G$ of the $z$-plane defines, in a natural way, a field of ...

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144 views

### Volume ratio of subsets of SO(n)

Let $\mu$ be the Haar measure on $SO(n)$. Consider the following subset:
$$
C_a=\left\{O\in SO(n):\sum_{i=1}^r\lambda_i\left(\sum_{j=1}^{n-r}O_{ij}^2\right)\leq a\right\}$$
where $\lambda_i>0,\sum_{...

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60 views

### Recovering the Weinstein Splitting Theorem for Poisson manifolds using the Local Splitting theorem for Lie algebroids

How can I recover the Weinstein Splitting Theorem for Poisson manifolds using the Local Splitting theorem for Lie algebroids? I am using the formulation of the theorem given in
Rui Loja Fernandes, ...

**2**

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**1**answer

102 views

### Are normal coordinates the same as Cartesian coordinates in flat space?

Let $\gamma_v$ be the unique maximal geodesic with initial conditions $\gamma_v(0)=p$ and $\gamma_v'(0)=v$ then the exponential map is defined by
$$\exp_p(v)=\gamma_v(1)$$
If we pick any orthonormal ...

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26 views

### Stable region of minimal hypersurfaces with finite Morse index

In this Inventiones Mathematicae paper, Fischer-Colbrie proved the following result (Proposition 1):
Proposition: Let $ M$ be a complete two-sided minimal surface in a three manifold $N$. Then if $M$...

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57 views

### Covariant derivative of the Monge-Ampere equation on Kähler manifolds

I am reading D. Joyce book "Compact manifolds with special holonomy" and I have some problems of understanding some computation on page 111, the first line in the proof of Proposition 5.4.6. More ...

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**1**answer

49 views

### Geodesic in half cone [closed]

Consider a right semicircular cone with height $h$ and radius $r$ given by $\mathcal{C}=\left\{\left(\frac{h-u}{h}r\cos(\theta),\frac{h-u}{h}r\sin(\theta),u\right)\,:\,u\in[0,h],\,\theta\in[0,\pi]\...

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217 views

### Geometry of the irrational torus

One of the motivations of diffeology is to study singular spaces such as the irrational torus. The irrational torus $T_α$ of slope $α∈R∖Q $ as a diffeological space is given by the quotient space $ R/(...

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132 views

### Extending Green's theorem from very special regions to more general regions

Green's theorem
Let $C$ be a positively oriented and consists of a finite union of disjoint,piecewise smooth simple closed curve in a plane, and let $D$ be the region bounded by $C$. If $P$ and $Q$ ...

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155 views

### Singular/Meromorphic maps into projective spaces

This may be a very basic question, so my apologies if that is the case.
But I was interested in having some examples of meromorphic (singular) maps into complex projective space from complex surfaces ...

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67 views

### Circle foliations not induced by circle actions on an compact orientable manifold

It is known that if we have an orientable fiber bundle $E\to B$, with fiber a circle $\mathbb{S}^1$, then it is a principal $SO(2)$-bundle. In other words, the fibers are spanned by the orbits of a ...

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32 views

### Integral scalar curvature of the submanifold

Let $(X,g)$ be a closed Riemanian manifold, and $Y\to X$ be an embeded submanifold. We denote by $N(Y)$ the tubular neighborhood of $Y$ and $Z=\partial N(Y)$.
On $TY$ we can calculate the scalar ...

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51 views

### Is the Frenet frame is independent of the choices of parameters?

I asked this question on StackExchange, but until now there is not any answer or hint. I hope I can get some help here.
When I am reading ''A course in differential geometry'' of Klingenberg, I ...

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67 views

### Airy stress, Beltrami stress and gauge fields

The following problem comes from the theory of elasticity, but reduces to a pure geometric problem. Consider a $d$-dimensional Riemannian manifold $(M,g)$ with boundary representing the intrinsic ...

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266 views

### Covariant derivative of determinant of the metric tensor

Let $(M,g)$ be a Riemannian manifold and $g$ the Riemannian metric in coordinates $g=g_{\alpha \beta}dx^{\alpha} \otimes dx^{\beta}$, where $x^{i}$ are local coordinates on $M$. Denote by $g^{\alpha \...

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129 views

### Nontrivial Gauss-Manin connection

Suppose $p: X \rightarrow S$ is a fiber bundle of smooth manifold, if the Gauss-Manin connection is nontrivial, could $p$ be trivial bundle as smooth manifold? Also, could $p$ be trivial bundle as ...

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370 views

### What does the torsion-free condition for a connection mean in terms of its horizontal bundle?

I must have read and re-read introductory differential geometry texts ten times over the past few years, but the "torsion free" condition remains completely unintuitive to me.
The aim of this ...

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**1**answer

123 views

### Weyl law for (non-semiclassical) Schrodinger operator

The Weyl law for a semiclassical Schrodinger operator
$$ A_h\ := \ -h^2\Delta+V(x) $$
on an $d$-dimensional complete Riemannian manifold $M$
says that the number $N(A_h,1)$ of eigenvalues of $A_h$ ...

**0**

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**1**answer

125 views

### Every complex vector bundle over the circle is trivial [closed]

Let $E \rightarrow S^1$ be a smooth complex vector bundle over $S^1$ (here complex means that the fibers have a vector space structure over $\mathbb{C}$).
Is it true that $E$ is necessarily trivial? ...

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**1**answer

185 views

### Reference Request: Uniformization Theorem

I would appreciate if someone could point me to some introductory literature/resources where I can learn about Poincare's Uniformization Theorem at a basic level.
Any good powerpoint notes, short ...

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12 views

### Calculate $\Phi ^{*}\omega $ for a given $\omega$ [migrated]

$\Phi : \mathbb{R}^{3}\rightarrow \mathbb{R}^{3}$
$(x,y,z) \rightarrow (xy,yz^{2},z^{3})$
Calculate $\Phi ^{*}\omega $ for :
i) $\omega= xdx\wedge dz - dx\wedge dy$
ii) $\omega= xdx\wedge dy \...

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46 views

### Spaces of sections of a holomorphic fiber bundle with specific normal bundles

There is a well-known fact (see for example HKLR) that if $p\colon Z\to \mathbb CP^1$ is a holomorphic fiber bundle admitting a holomorphic section $s\colon \mathbb CP^1\to Z$ such that $s^*N\cong \...

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**2**answers

188 views

### Lifting sections of a projective bundle to a vector bundle

Let $E\to M$ be a smooth $\mathbb{K} = \mathbb{R}, \mathbb{C}$ - vector bundle over a possibly non-compact connected manifold $M$. Denote by $\mathbb{P}(E) \to M$ its projectivization, which is ...

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91 views

### Volume growth of balls in manifolds with bounded geometry

Suppose $(M,g)$ is a Riemannian manifold. Let us say that $M$ has bounded geometry if its injectivity radius is uniformly bounded below by a constant $\epsilon>0$, and the curvature tensor and all ...

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99 views

### Plane projection of Geodesics (Inverse view)

Maybe this question is so clear or maybe it is not exact. It is because of my very little knowledge of differential geometry. I am reading some material in this field and I got a question which seems ...

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**1**answer

143 views

### Characterisation of Sobolev Spaces on manifolds of bounded geometry via geodesic coordinates

I have a reference request concerning equivalent norms on Sobolev Spaces on manifolds of bounded geometry. This may be obvious to the experts but I am not working in the field and only want to use ...

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173 views

### Can this integral be made nonpositive?

Let $M^2 \subset \mathbb{S}^3$ be a closed and orientable embedded (and minimal, if important) surface. Choose a unit normal vector field $\eta: M \to \mathbb{S}^3$ along $M$ and a point $p_0 \in \...

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54 views

### The idealizer of the space of vector fields with vanishing divergence

The space of smooth vector fields on a manifold is counted as a Lie algebra with the usual Lie bracket structure.
Is there a Riemannian manifold of dimension at least $2$ which satisfies either of ...

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63 views

### Computation of equivariant 3 form

I want to how an equivariant 2-form and equivariant 3- form look like i,e.,
Let M be a complex Manifold say $ S^4 \subset C^3$ and a compact lie group $S^1$ acting on it via the action $\exp{i\theta}....

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**1**answer

122 views

### An integral of the Hodge-Neumann Laplacian on a Riemannian manifold

Background
Let $M$ be a compact, oriented Riemannian manifold with boundary, with $\text{vol}$ the Riemannian volume form. Let $\nu$ be the outwards unit normal vector field on $\partial M$ and $\nu^\...

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**1**answer

231 views

### Can the number of solutions to a system of PDEs be bounded using the characteristic variety?

I've recently come across a system of PDEs which I'd like to understand better. The particular system I'm interested in locally solves for a 2-dimensional Riemannian metric as the Hessian of a ...

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**1**answer

113 views

### Existence of nonvanishing Killing field

Let $(M,g)$ be a closed Riemannian manifold.
Q Is there any research about the existence of nonvanishing Killing field, especially the nontrivial example.

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94 views

### Alternative Lie bracket on $\chi^{\infty}(M)$ where $M$ is a Riemannian manifold with a symplectic structure

Let $(M,\omega, g)$ be a Riemannian symplectic manifold. The Poisson bracket of two functions $f,g$ is denoted by $\{f,g\}$. The Hamiltonian vector field and gradient vector field ...

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85 views

### Metrics with prescribed Levi-Civita connection

My question involves the symmetries of a (pseudo)-Riemannian metric preserving the Levi-Civita connection (LCC), its unique torsion-free metric connection. For a basic example, one notes that the ...

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**1**answer

64 views

### How to show if $X$ is Killing field then it is tangent to the geodesic spheres centred at a point $p$?

Let $M$ be a Riemannianiam manifold with Levi-Civita connection and $X $ be a smooth vector field on $M$. Let $\phi : (-\epsilon, \epsilon) × V \to M$ be the local flow of $X$ in $M$.
Problem is- if $...

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47 views

### Is Colding-Minicozzi entropy continuous w.r.t. $C^\infty$ convergenge?

For an hypersurface $\Sigma^n \subseteq \mathbb{R}^{n+1}$ the entropy introduced by Colding and Minicozzi (see their paper) is defined as
$$
\lambda(\Sigma) := \sup_{x_0 \in \mathbb{R}^{n+1} \\ t_0 \...

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98 views

### Gauge invariance of Chern-Simons functional integral for a 3-manifold with boundary

I am trying to understand how the functional integral for Chern-Simons theory for a possibly non-compact 3-manifold with boundary is made gauge invariant.
For a compact 3-manifold, $M$, without ...