All Questions
Tagged with reference-request dg.differential-geometry
800 questions
3
votes
2
answers
348
views
Request for some references exploring the connections of Riemann surfaces with medical imaging
I'd like to know some references for a beginner who has basic background in Riemann surfaces and differential geometry, and would like to start learning/working on more applied areas, medical imaging/...
- 1,467
3
votes
3
answers
2k
views
How do we use an Ehresmann connection to define a semispray?
Let $M$ be a differentiable manifold, let $TM$ be its tangent bundle, and consider $TTM$, the double tangent bundle.
Let $V \subseteq TTM$ denote the vertical subbundle, which is determined in a ...
- 8,512
3
votes
2
answers
321
views
parallel transport along $W^{1,2}$-curves
Let $c$ be a $W^{1,2}$-curve into a (compact Riemannian) manifold $Q,$ defined on some open interval $I$. Let $t_0\in I$ and $\xi_0\in T_{c(t_0)}Q$ be arbitrary. I am looking for a citeable reference ...
- 2,935
3
votes
1
answer
160
views
Total curvature of a Caustic
I am looking for a nudge in the right direction on the derivation of a formula for the Total Curvature of the Caustics to a manifold (a caustic is a planar family of curves reflected by a manifold).
...
- 173
3
votes
1
answer
280
views
About the metric and embedding of sphere
Let $S^2$ be the $2$-dimensional sphere with a metric $g$.
Q:
Can we or how to find a smooth map $f:S^2\to \mathbb R^3$, such that
(1) $f$ is diffeomorphic to its image $Im(g)=:M$,
(2) $M$ ...
- 1,915
3
votes
1
answer
466
views
Finding global sections of a sheaf of sets using (some kind of) sheaf cohomology?
Let $X$ be a compact manifold, say, and $G$ a Lie group, and $H$ a closed Lie subgroup such that $M \cong G/H$ is a homogeneous space. (For my purposes, $X$ and $M$ would be a smooth projective ...
- 1,763
3
votes
1
answer
195
views
Quotient by freely acting group on Banach manifold
I have a Banach manifold $\mathcal{M}$ and I have a Lie group $G$, that is finite dimensional, such that $G$ acts freely on $\mathcal{M}$. I would like to know if $\mathcal{M} / G$ is a Banach ...
- 431
3
votes
1
answer
232
views
Vector fields $X$ and $Y$ commute on a closed set $K$. Do there exist commuting $\tilde X,\tilde Y$ with $\tilde X=X$ and $\tilde Y=Y$ on $K$?
I have a nice research idea whose proof hinges on the following question
Suppose $X_p$ and $Y_p$ are vector fields in $\mathbb{R}^3$ with $[X,Y]_p=0$ for all $p$ in some closed set $K\subset\mathbb{R}...
- 133
3
votes
1
answer
270
views
survey paper on the construction of hyperbolic manifolds
Is there a good survey paper which discusses the common ways of building hyperbolic $n$-manifolds?
- 101
3
votes
1
answer
338
views
'Unitary' charts on odd-dimensional spheres
Consider the odd-dimensional sphere $S^{2n+1} \subset \mathbb{C}^{n+1}$. One may talk variously about its structure as a contact, CR or Einstein-Sasaki manifold, but I'm looking for some specific down-...
- 35.5k
3
votes
1
answer
459
views
Are Carnot groups (as Carnot Caratheodory metric spaces) doubling?
I need to use the Lebesgue differentiation theorem for doubling metric measure spaces and was wondering if Carnot groups are doubling. If yes, is there any reference you can point me to? Thank you.
- 520
3
votes
1
answer
228
views
Exposition of the Calabi complex
I am interested in a complex derived by Eugenio Calabi in his article "On compact Riemannian manifolds with constant curvature".
The complex is referenced as "Calabi complex" in various citing ...
- 5,327
3
votes
2
answers
347
views
Direct calculation of the Fisher distance via Riemannian geodesics
I'm looking for a reference for a direct calculation of the Fisher distance (to avoid overloading the term "metric") $d_F(x,y) := 2 \cos^{-1} \sum_i \sqrt{x_i y_i}$ as the geodesic distance ...
- 15.4k
3
votes
1
answer
455
views
Identity Theorem for Real-Analytic Hypersurfaces
There's an interesting statement it seems I can prove, but I can't find any references for it, which makes me suspicious of it. So, could someone verify that the statement is correct/incorrect or ...
- 343
3
votes
2
answers
1k
views
Reference for homogeneous spaces
I am a graduate student of differential geometry.
I would like to get an overview over the way, how results are usually obtained for homogeneous spaces by Lie algebraic methods. By definition a ...
- 79
3
votes
1
answer
81
views
Estimating the Size of an Approximating Polyline
let $\gamma(s) = \left(x(s),y(s)\right), s\in[0,1]; \gamma'(s) = 1$ be a length-parameterized curve in the plane, with finite and strictly positive curvature.
Questions:
is it possible to estimate ...
- 13.2k
3
votes
1
answer
215
views
How many second-order PDEs can be obtained from a contact EDS?
Let $(M,[\theta])$ be a contact manifold, $\dim M=2n+1$, and denote by $\mathcal{I}^\theta$ the differential ideal generated by the contact form $\theta$.
An exterior differential system on $M$ of ...
- 2,527
3
votes
1
answer
632
views
Intuition behind the Duistermaat-Guillemin version of Weyl's law
The theorem in question (see this paper), after a modification by Ivrii (see this paper) states the following:
Let $(M, g)$ be a compact Riemannian manifold of dimension $n \geq 2$. Assume that the ...
- 31
3
votes
1
answer
255
views
Norm on space of metrics
I recently heard a differential geometry talk where the speaker constructed a one-parameter family of metrics $g(t)$ on a smooth manifold and said that $g(t)$ is real analytic in the Banach space $BC(...
- 51
3
votes
1
answer
344
views
derivative of the adiabatic limit of the eta invariant
To ask my question I have to write down the setup. Basically the setup is the adiabatic limit of the reduced eta invariant of Dirac operator associated to the submersion metric and connection. So if ...
- 1,173
3
votes
1
answer
203
views
Cohomology of the complex of differential forms with Schwartz coefficients
Let $U$ be an open manifold (say an open subset of $\mathbb{R}^n$ for simplicity). Denote by $\mathscr{S}(U)$ the space of Schwartz functions on $U$. Schwartz functions are defined as usual to be ...
- 1,374
3
votes
1
answer
313
views
Infinite number of closed geodesics on distorted sphere
I would appreciate a reference to support this statement that
appears under the Geodesic entry of the
CRC Encyclopedia of Mathematics:
"no matter how badly a sphere is distorted,
there exists an ...
- 151k
3
votes
1
answer
369
views
Closed manifolds of nonnegative curvature operator are symmetric spaces
In an online webinar, I heard (not directly) the statement that (closed) manifolds of nonnegative curvature operator $\mathcal{R}\geq 0$ are symmetric spaces. Is this a valid theorem? Any reference ...
- 4,195
3
votes
1
answer
190
views
Laplace eigenfunction on a polygonal domain symmetric about an axis
Consider a polygon $\Omega \subseteq \mathbb{R}^2$, and let us consider the usual Laplacian operator $\Delta = \partial_x^2 + \partial_y^2$, with Dirichlet boundary conditions. My question comes from ...
- 33
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
3
votes
1
answer
239
views
What is the curved version of the Tits fibration for $G_2$?
Let
$\require{AMScd}$
\begin{CD}
G_2/(P_1\cap P_2) @= G_2/(P_1\cap P_2)=:\mathbb{I}\\
@V \lambda V V @VV \pi V\\
\mathbb{Q}_5:=G_2/P_1 && G_2/P_2=:\mathbb{N}_5
\end{CD}
be the Tits ...
- 2,527
3
votes
1
answer
284
views
Long time existence of Ricci flow on compact surfaces of negative curvature
Is there a long time existence for the Ricci flow on compact negatively curved surfaces? I just read that the normalized Ricci flow has a long time solution converging to a metric of constant negative ...
- 39
3
votes
1
answer
524
views
Kahler-Einstein metrics on Toric manifolds are Torus-invariant?
let $(M,\omega)$ be a Kahler-Einstein toric manifold of complex dimension $m$. By toric manifold i mean a manifold that has an open dense subset $X$ biholomorphic to an algebraic torus $\mathbb{T}^{m}...
- 1,727
3
votes
2
answers
597
views
Isometric Immersion of $S^1\to M$
$M$ be any Riemannian manifold, and $S^1$ is a circle.
We can give Manifold structure to $C^\infty(S^1, M)$ modeled on nuclear frechet space.
Take $Imm(S^1, M):\{f\in C^\infty(S^1,M): f \text{ is an ...
- 541
3
votes
1
answer
1k
views
Friedrichs mollifiers and Sobolev spaces
$\renewcommand{\epsilon}{\varepsilon}$The following is from John Roe's book Elliptic operators, topology and asymptotic methods. $S$ is a vector bundle on a compact manifold $M$, but I think for my ...
- 1,141
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 ...
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
3
votes
1
answer
195
views
Constructing a Hamiltonian $S^1$-action on a neighborhood of a symplectic divisor
Let $M^{2n}$ be a symplectic manifold and let $M^{2n-2}$ be a symplectic submanifold. How to construct a non-trivial Hamiltonian $S^1$-action on $M^{2n-2}$ on a small neighborhood of $M^{2n-2}$, that ...
- 14.3k
3
votes
1
answer
333
views
Elementary question: Curvature change under Complexified Gauge Transformation
Forgive me for this elementary question.
Let $E$ be a holomorphic vector bundle over a Riemann surface $M$ equipped with a Hermitian metric. Let $\nabla$ be the compatible connection on $E$ amd $g$ ...
- 297
3
votes
1
answer
560
views
Prescribing an induced metric
We know that, if we have a surface $z=f(x,y)$ with Euclidean space being ambient manifold, the induced metric is as follows (in matrix form):
$$g=\begin{bmatrix}
1+\left ( \frac{\partial f(x,y)}{\...
- 267
3
votes
1
answer
205
views
Reference: Finsler Derivative?
On the wikipedia page "Generalizations of derivative" the author mentions: " in Finsler geometry, one studies spaces which look locally like Banach spaces. Thus one might want a derivative with some ...
- 5,405
3
votes
1
answer
154
views
Regularity of maps in algebraic topology for manifolds
Let $M$ be a $n$ manifold such that $\pi_k(M)$ is non trivial. What can we expect about the regularity of a representant $f:S^k\rightarrow M$ of a non-trivial cycle? For example, if $M$ is a manifold ...
- 1,616
3
votes
1
answer
432
views
Is there a characterization of Riemannian manifolds that split off two factors?
Some Riemannian manifolds are expressed as a product manifold. Recently, I have read two articles about space-times. In both articles, the authors prove that a Riemannian manifold $\bar{M}^n$ is ...
- 422
3
votes
1
answer
300
views
Different ways of defining the Chern character of a complex
Consider a finite complex $E$ of (holomorphic) vector bundles on a (complex) manifold $X$, i.e, the complex is of the form
$$
0 \to E_N \to E_{N-1} \to \dots \to {E_0} \to 0,
$$
where the bundles are ...
- 1,457
3
votes
1
answer
286
views
Kähler metric of constant scalar curvature with positive bisectional curvature is Kähler-Einstein
Suppose $\omega$ is a Kähler metric of constant scalar curvature with positive bisectional curvature, how to prove $\omega$ is Kähler-Einstein?
I was told that we can use the following method:
Step ...
- 79
3
votes
1
answer
1k
views
Prerequisites for reading characteristic classes
Can some one tell me what are the prerequisites for learning characteristic classes as they are in book Foundations of Differential geometry by Kobayashi and Nomizu.
I only read first two chapters of ...
- 6,023
3
votes
3
answers
908
views
Good exposition of "Calabi ansatz"
As far as I understand, Calabi ansatz is (in particular) a way to produce Kähler metrics on total spaces of line bundles (or their disk subbudles) over Kähler manifolds of the following form:
Calabi ...
- 14.3k
3
votes
1
answer
292
views
Existence of Simple Closed Straightest Geodesics
There are at least three distinct simple closed quasigeodesics on convex polyhedra [Mat. Sb. (N.S.), 1949, 25(67) :2, 275–306 Quasi-geodesic lines on a convex surface Pogorelov].
Is the same true ...
- 133
3
votes
1
answer
198
views
Poincaré constant for $L^2$-differential-forms on a submanifold of $\mathbb R^n$ with Lipschitz boundary
Let $M \subset \mathbb R^n$ be a submanifold of euclidean space whose boundary is locally a Lipschitz graph. Let $\omega \in L^2\Lambda^k(M)$ be a differential form with square-integrable coefficients....
- 5,327
3
votes
1
answer
277
views
A (non-Kahler) metric on projectivised vector bundles
Given a hermitian holomorphic vector bundle (E, h) on a complex manifold-with-a-metric (X,g), then consider the following (natural) construction of a metric on the total space of $\mathbb{P}(E)$ : ...
- 3,383
3
votes
0
answers
76
views
Bow lemma with angles
First, let me recall the statement of the bow lemma.
Let $\gamma_1: [a,b] \to \mathbb{R}^2$ and $\gamma_2: [a,b] \to \mathbb{R}^2$ be two smooth unit-speed curves.
Assume $\gamma_1$ and its chord ...
- 45k
3
votes
0
answers
153
views
Quasimode construction on a compact Riemannian manifold
Let $M$ be a closed Riemannian manifold, $\Delta$ be the usual Laplace-Betrami operator on $M$ and $\gamma : [0, L] \to M$ be a stable elliptic periodic geodesic of length $L$. I have heard in several ...
- 131
3
votes
0
answers
239
views
Critical points up to smooth homotopy
Let $M$ and $N $ be closed connected smooth manifolds of dimension $n$.
Let $f: M\rightarrow N$ be a smooth function and not null-homotopic. Is there a smooth homotopy $H: [0,1]\times M\rightarrow N$ ...
- 223
3
votes
0
answers
233
views
A bridge between the algebraic / differential geometry of $\frak{sl}_2(\mathbb{C})$ and the Sheffer-Appell calculus and combinatorics
In "Four examples of Beilinson-Bernstein localization", Anna Romanov introduces the basis
$m_k = \frac{(-1)^k}{k!} \partial^k \delta $
on p. 9, where $\partial$ is a partial derivative and $\...
- 10.5k
3
votes
0
answers
109
views
"Practical" references on mapping spaces as infinite-dimensional manifolds
I am studying spaces of the form $C^{k}(\mathcal{M},\mathcal{N})$ between manifolds ($k=\infty$ allowed) and I am looking for extensive references, especially analysing their topology and smooth ...
- 1,171