All Questions
Tagged with reference-request dg.differential-geometry
800 questions
2
votes
1
answer
149
views
Surveys/monographs on the vortex filament equation
Where can I find surveys on the mathematical aspects of the vortex filament equation?
In particular, I'm interested in the following topics:
physical motivation;
notion of solutions and ...
- 277
3
votes
0
answers
336
views
Understanding Calabi's conjecture proof: What is it meant by the logarithm of a differential form?
I'm reading several books and articles concerning Yau's proof of the Calabi conjecture. I want to have a deep understading of how and why such proof actually works, but most articles are aimed at ...
- 153
7
votes
1
answer
612
views
Foliation with trivial leaf holonomy
In 1960, R. Hermann showed the following:
Theorem Let $M$ be a manifold with a foliation $F$ and a bundle-like metric, if all leaves are compact and the holonomy group of each leaf is trivial, then $...
- 1,915
5
votes
1
answer
291
views
Isotropy subgroupoid of a regular Lie groupoid
Let $(G\rightrightarrows M)$ be a Lie groupoid (i.e. a groupoid with source map $s$ and target map $t$ such that $G,M$ are smooth manifolds and the structural maps are all smooth (and $s$,$t$ are ...
- 2,388
2
votes
0
answers
269
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 $Q$ ...
- 151
10
votes
2
answers
938
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 $A_h$ ...
- 658
6
votes
2
answers
706
views
Reference request: uniformization theorem
I would appreciate if someone could point me to some introductory literature/resources where I can learn about Poincaré's uniformization theorem at a basic level.
Any good powerpoint notes, short ...
- 63
5
votes
1
answer
178
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 ...
- 4,784
6
votes
1
answer
342
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 ...
- 6,001
5
votes
1
answer
594
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.
- 1,915
4
votes
0
answers
244
views
Harmonic maps into de Sitter Space
I am looking some references on the existence and non-existence of (space-like) harmonic maps solving the Dirichlet into the de-Sitter space.
More precisely: Let, for $n\geq 3$,
$$dS^n=\{ u\in \...
- 914
1
vote
1
answer
840
views
Reference request: Gauge theory [closed]
What are some good introductory texts to gauge theory? I have some basic differential geometry knowledge, but I don’t know any algebraic geometry.
Also, as a side question, what intuitively is a ...
- 2,069
0
votes
1
answer
166
views
Non-trivial foliation (excluding the Reeb foliation) [closed]
Let $M$ be a closed oriented manifold, an oriented foliation $F$ is said non-trivial, if $F$ is not fibration of $M$, i.e. there does not exist a closed manifold $B$, such that $M\overset{F}{\to} B$.
...
- 1,915
4
votes
0
answers
129
views
Coordinate-free B.Feix's construction of a hyperkähler metric
In the 2001's paper 'Hyperkähler metrics on cotangent bundles' B.Feix gives a construction of a hyperkähler metric on a neighbourhood of zero section in $T^*X$ where $X$ is a real analytic Kähler ...
- 2,305
4
votes
2
answers
405
views
Gaussian upper heat kernel bounds on closed Riemannian manifolds
Let $M$ be a closed Riemannian manifold, and let $h(t, x, y)$ denote the heat kernel on $M$. We know that there exists short time upper Gaussian heat kernel bounds of the following kind:
$$
h(t, x, y) ...
- 1,407
4
votes
0
answers
101
views
Terminology for a foliation that is only tangentially smooth
I'd like to get some information and references, starting with a name, for the following quite common situation, for a smooth (i.e. $C^\infty$) $n$-manifold $M$. A partition $\mathcal{L}$ of $M$ is ...
- 60.5k
3
votes
0
answers
312
views
Foliated vector bundle and basic connection
Let $(M,F)$ be a Riemannian foliation, i.e. there is a metric of $TM$ such that $g$ is bundle-like (locally $g_Q=g_{ij}(y)\,dy^i\otimes dy^j$ for a foliated chart $(x,y)$ and $Q=TM/F\cong F^\perp$).
...
- 1,915
15
votes
1
answer
612
views
Is the subgroup $\mathrm{Diff}(M,S)$ of $\mathrm{Diff}(M)$ a Lie subgroup?
Denote by $\mathrm{Diff}(M)$ the Lie group of smooth diffeomorphisms on a compact smooth manifold. Its Lie algebra can be viewed as the Lie algebra $\mathfrak X(M)$ of vector fields on $M$. Now, given ...
- 2,789
2
votes
0
answers
255
views
Sobolev Multiplication on non-compact manifold
We know that for a compact Riemannian $n$-dim manifold $(M,g)$(the boundary could be nonempty), the Sobolev Multiplication Theorem states that $L^p_k\times L^q_l⟶L^r_m$, where $1/r−m/n>1/p−k/m+1/...
- 1,915
10
votes
1
answer
445
views
conditions for long geodesics without self-intersections
Consider a Riemannian manifold $\Sigma$ of dimension two homeomorphic to a torus. When is there a non-closed geodesic on $\Sigma$ which does not intersect itself — are there reasonable necessary or ...
- 6,891
1
vote
2
answers
1k
views
Reference on Complex Geometry
For the preparation of a complex geometry lecture I am looking for a good literature. I already have standard literature like Huybrechts "Complex Geometry. An Introduction" and I am also using it. But ...
- 39
4
votes
1
answer
129
views
Holomorphic Poisson structures on $C P^{n-1}$ and homogeneous Poisson structures on $C^n$
Is it correct that any holomorphic Poisson structure on $C P^{n-1}$ can be lifted to a homogeneous Poisson structure on $C^n$? By homogeneous I mean a quadratic Poisson structure of the form $\{z_i,...
3
votes
1
answer
628
views
Local Sobolev embedding on complete Riemannian manifold
Let $(M,g)$ be a complete Riemannian $m$-manifold, with bounded geometry and $m\geq2$. Suppose $Ric\geq(n-1)\kappa$. Let $B_p(r)$ be a geodesic open ball.
Q Can we find a constant $C=C(\kappa,r,m)$(...
- 1,915
0
votes
0
answers
139
views
(Semi-)Riemannian geometry for working PDE analysts
What is a good reference on (semi-)Riemannian geometry written for PDE analysts (that is, with main focus on analytical problems and approaches)?
The closest thing I know to this, are two books by ...
user60665
0
votes
1
answer
119
views
Analytic approach to geodesic connectedness in Semi-Riemannian manifolds
Can you point out a reference (or references) that deal with analytical methods (rather than methods from differential geometry) for the study of geodesic connectedness on Semi-Riemannian manifolds?
user60665
3
votes
0
answers
114
views
Jacobian of the action of a matrix on a Grassmannian
I'm looking for a reference concerning a calculation found in Furstenberg's 1963 paper "Non-commuting random products".
Lemma 8.8 of this paper states that if one takes a $d\times d$ invertible real ...
- 4,304
1
vote
0
answers
68
views
Curvature of projection function onto a smooth curve
Suppose we have a smooth curve $C$ lying in $\mathbb{R}^2$, and let us consider the orthogonal projection function $P_C(x)$ onto the curve, described by
$$P_C(x) = argmin_{y \in C} \Vert x - y \Vert$$...
- 141
5
votes
0
answers
307
views
Gradient estimate for Poisson equation on manifold
In Gilbarg-Trudinger's book 'Elliptic Partial Differential Equations of Second order', the maximum principle is used to derive the following gradient estimates for Poisson equations on Euclidean ...
- 2,789
7
votes
1
answer
372
views
Theory of surfaces in $\mathbb{R}^3$ as level sets
Is there a book that treats the classical theory of surfaces in $\mathbb{R}^3$ from the point of view of level sets of a function? I seem to remember someone telling me that such a book exists, but I ...
- 7,197
17
votes
2
answers
2k
views
The Lefschetz operator
Let $\omega=\sum_{i=1}^n dx_i\wedge dy_i\in\bigwedge^2(\mathbb{R}^{2n})^*$ be a standard symplectic form. The following result is due to Lefschetz:
For $k\leq n$, the Lefschetz operator
$L^{n-k}:\...
- 28k
3
votes
2
answers
782
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 ...
8
votes
2
answers
238
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 ...
- 2,305
14
votes
1
answer
905
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 ...
- 45k
10
votes
1
answer
403
views
Positive Ricci curvature on fiber bundles
My advisor and I are working on Ricci curvature and an anonymous referee pointed out the following conjecture:
Let $F\hookrightarrow M\stackrel{\pi}{\to}B$ be a fiber bundle from a compact manifold ...
- 1,774
6
votes
1
answer
870
views
Physical (GR) Differential Geometry?
I am looking for problem lists or books which contain open problems in the area of mathematics motivated by physics. Ideally, I am looking for questions asking about which reduce to some calculation ...
- 191
3
votes
1
answer
314
views
Foliation with a compact leaf
Let $M$ be a closed oriented manifold, and $F$ be a fixed foliation of $M$. We assume the dimension and codimension of $F$ are both greater than $1$.
Q Under what condition, we can say that $F$ ...
- 1,915
2
votes
0
answers
66
views
One-parameter group of nonvanishing vector field
Let $M$ be a smooth manifold( if necessary one can assume it is closed), $V$ be a non-vanishing vector field of $TM$.
Q: Under what condition, we can say that $\overline{\exp(tV)}$, i.e. the closure ...
- 1,915
3
votes
0
answers
129
views
Differential operators on a compact Lie group associated to bracket-generating sets
Let $G$ be a compact connected Lie group of dimension $m$ with Lie algebra $\mathfrak g$.
Let $\{X_1,\dots,X_h\}$ be a linearly independent set of $\mathfrak g$.
Assume that $\{X_1,\dots,X_h\}$ is ...
- 2,446
2
votes
0
answers
126
views
The complex Clifford algebra
If $(E,g,w)$ is a vector space $E$ with a metric $g$ and a symplectic form $w$; then we can define the complex parts $(1,0)$ and $(0,1)$, so that the complex Clifford algebra is: $$e_1 . f_1 + f_1 . ...
- 21
1
vote
0
answers
96
views
The hermitian Einstein manifolds
I take an hermitian manifold $(M,g,J)$ and I define from the riemannian curvature $R(X,Y)$ as a $2$-form:
$$
Ricc(J)= \sum_i R(J e_i,e_i)
$$
with $(e_i)$ an orthonormal basis of the tangent.
$$
2R(J)=...
- 187
5
votes
1
answer
169
views
Quick question on the constants involved in heat kernel upper bounds
Let $M$ be a compact Riemannian manifold without boundary, and let $\Delta$ be the Laplace-Beltrami operator on $M$. It is known that for small $t$, let's say, $0< t < t_0(M, g)$, the heat ...
- 51
4
votes
0
answers
152
views
Gromov–Hausdorff distance between Riemannian manifolds
Let $(M,g)$ be a Riemannian manifold; for the sake of simplicity we assume that its group of isometries is trivial. If we consider the same manifold equipped with another metric $g'$, what is the ...
- 6,891
10
votes
2
answers
480
views
Geometric description of a certain sphere bundle
It appears to be a standard fact in topology that $\mathbb{C}\mathbb{P}^2\#-\mathbb{C}\mathbb{P}^2$ has a structure of a $\mathbb{S}^2$ bundle over $\mathbb{S}^2$. Is there a nice geometric ...
- 6,891
6
votes
0
answers
355
views
Higher order variations of Riemannian geodesics
Consider a mapping $\Gamma$ from the Euclidean plane or an open subset to a Riemannian manifold $M$ so that each $\Gamma(s,\cdot)$ is a geodesic.
There is a well established theory of the first order ...
- 8,086
4
votes
0
answers
116
views
$\ell_p$ geodesic distance on smooth Riemannian manifold and Logarithmic Sobolev Inequalities
Bear with me, I'm not a professional geometer. Recently, I've been studying Logarithmic Sobolev Inequalities (LSI) for probability distributions on manifolds (e.g as done in works of Bobkvo et al. ...
- 6,853
16
votes
5
answers
2k
views
Reference request: Recovering a Riemannian metric from the distance function
Let $M = (M, g)$ be a Riemannian manifold, and let $p \in M$.
Writing $d$ for the geodesic distance in $M$, there is a function
$$
d(-, p)^2 : M \to \mathbb{R}.
$$
This function is smooth near $p$. ...
- 27.7k
4
votes
0
answers
343
views
Diffeomorphism group action on the space of embeddings
Let $S$ and $M$ be two finite-dimensional smooth manifolds with $\dim S\le \dim M$. Then it is known (e.g.Kriegl-Michor's book) that the set $\mathrm{Emb}(S, M)$ of all smooth embeddings $S\to M$ is ...
- 2,789
2
votes
2
answers
1k
views
The existence of length-minimizing path between two points in a Riemannian manifold with boundary
Let $(M^n,g)$ be a Riemannian manifold with non-empty smooth boundary $\partial M$. For any two points $x,y\in M$, the distance between $x$ and $y$ may be defined as
$$ d(x,y)=\inf_\gamma Length(\...
3
votes
3
answers
1k
views
Lie algebra bundle associated to a Lie group bundle
I was reading the paper Non abelian Differentiable gerbes (page 24) and came across notion of Lie algebra bundles associated to a Lie group bundle.
I am not comfortable with these notions and google ...
- 6,023
2
votes
1
answer
239
views
Projection of a ball in the ambient space to a manifold
Let $B_h (x)$ be the ball of radius $0<h \ll 1$ centered at $x\in \mathbb{R}^d$.
Let $I=[0,1]^{d-1}$ be the unit cube in $\mathbb{R}^{d-1}$, and let $f:I \to \mathbb{R}$ be a $C^2$ function. Then $...
- 3,574