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Questions tagged [dg.differential-geometry]

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

1
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1answer
13 views

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
votes
1answer
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 $\check{H}^2(X,\mathbb{R})$,...
6
votes
0answers
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 $s\in \Gamma(E)$. ...
3
votes
0answers
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(\...
-4
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0answers
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: $$ \...
6
votes
1answer
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,......
0
votes
0answers
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^...
2
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0answers
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 ...
4
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0answers
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. ...
4
votes
0answers
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 $S_n$? ...
3
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0answers
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,......
5
votes
1answer
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 ...
-2
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0answers
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)}\\ &...
5
votes
1answer
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 ...
5
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0answers
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_{...
3
votes
0answers
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
votes
1answer
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 ...
1
vote
0answers
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 $M$...
2
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0answers
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 ...
0
votes
1answer
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]\...
3
votes
0answers
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/(...
1
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0answers
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 $Q$ ...
0
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0answers
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 ...
7
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0answers
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 ...
1
vote
0answers
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 ...
1
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0answers
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 ...
4
votes
0answers
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 ...
3
votes
3answers
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 \...
1
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0answers
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 ...
12
votes
3answers
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 ...
6
votes
1answer
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
votes
1answer
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? ...
2
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1answer
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 ...
0
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0answers
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 \...
3
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0answers
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 \...
4
votes
2answers
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 ...
0
votes
0answers
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 ...
4
votes
0answers
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 ...
5
votes
1answer
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 ...
4
votes
0answers
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 \...
0
votes
0answers
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 ...
2
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0answers
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}....
3
votes
1answer
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^\...
6
votes
1answer
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 ...
3
votes
1answer
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.
1
vote
0answers
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 ...
3
votes
0answers
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 ...
0
votes
1answer
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 $...
2
votes
0answers
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 \...
4
votes
0answers
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 ...