All Questions
Tagged with reference-request dg.differential-geometry
800 questions
1
vote
1
answer
755
views
Tensor analysis/Differential forms outside physics
There are many "geometric systems" like tensor analysis or differential forms calculus, which more or less different perspectives onto the same abstract relations.
Most applications are physical, ...
2
votes
0
answers
260
views
Perturbation of Morse function at a critical point
I recently learned from a knowledgeable person that for a Morse function $f: M \to R$ with a critical point $x_0$, one can perturb $f$ in such a fashion that the new function has the same critical ...
- 1,211
23
votes
3
answers
3k
views
Hsiung on the Complex Structure of $S^6$
In 1986 C. C. Hsiung published a paper "Nonexistence of a Complex Structure on the Six-Sphere" and in 1995 he even wrote a monograph "Almost Complex and Complex Structures" to further elaborate on his ...
- 329
36
votes
2
answers
5k
views
Kervaire invariant: Why dimension 126 especially difficult?
Is there any resource that might help non-experts gains some understanding of why
the Kervaire invariant problem remains open now only in dimension $126$? ($126 =2^7-2=2^{j+1}-2$;
whether $\theta_j=\...
- 151k
1
vote
0
answers
371
views
Simple development of simple curve on a cone
Let $\Lambda$ be a cone with apex $a$ and apex angle $\alpha$. Draw a simple (non-self-intersecting)
curve $C=(x,y)$ on $\Lambda$, and then develop it to a curve
$\overline{C}$ on a plane by rolling $...
- 151k
8
votes
2
answers
994
views
Homotopy invariance of vector bundles by parallel transport: reference needed for my students.
Let $M$ be a smooth manifold and $V \to [0,1] \times M$ be a smooth vector bundle. The homotopy invariance states that the restrictions $V_0$ and $V_1$ to the bottom and top of the cylinder are ...
- 20.9k
33
votes
8
answers
9k
views
"Modern" proof for the Baker-Campbell-Hausdorff formula
Does someone has a reference to a modern proof of the Baker-Campbell-Hausdorff formula?
All proofs I have ever seen are related only to matrix Lie groups / Lie algebras and
are not at all geometric (...
- 2,074
4
votes
3
answers
2k
views
book on PDE on manifolds
let $M$ be a Riemannian manifold and $\alpha$ be any some unknown form on $M$. I am interested in solutions or some references of the equation of type $(d + \delta) \alpha = 0$ where $\delta$ is the ...
- 89
6
votes
2
answers
3k
views
References for the Poincaré-Cartan forms
Hello, everybody. I'm looking for some reference about the Poincaré-Cartan form, I do not know how it is defined, I just know that it is used in Lagrangian mechanics but I have not found any ...
- 463
6
votes
3
answers
889
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Reference request: embedded Morse theory
For references of embedded Morse theory, or so-called relative morse theory of a pair, I have found R.W. Sharpe's "Total Absolute Curvature and Embedded Morse numbers" totally not helpful. For further ...
- 2,274
7
votes
4
answers
3k
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Levy-Gromov Isoperimetric Inequality
In his paper "Paul Levy's Isoperimetric Inequality", Gromov gives the following isoperimetric inequality:
Let $V$ be a closed $(n+1)$-dimensional Riemannian Manifold with $\mathrm{Ric}(V) \geq n \...
- 1,123
7
votes
1
answer
723
views
Complex manifolds with corner?
I was reading Dominic Joycee article on Manifold with corner. He talk about manifold with corner modeled over $[0,\infty)^k\times \mathbb R^{n-k}$ for some $k\leq n$. From here I moved to Melrose ...
- 227
1
vote
1
answer
259
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$L^2$-de-Rham complex on Lipschitz domains has smooth harmonic forms?
I would like to know for which choice of boundary conditions the title statement is true.
Let $\Omega$ be a bounded Lipschitz domain in $\mathbb R^n$, for which we regard the $L^2$-de-Rham complex.
...
- 5,327
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
5
votes
0
answers
1k
views
"The famous Lusternik-Schnirelmann Theorem of the Three Closed Geodesics"
The title is a quote from p.256 of Wilhelm Klingenberg's 1995
Riemannian Geometry (Google Books link):
Every surface homeomorphic to a sphere $\mathbb{S}^2$ has three distinct, simple, closed ...
- 151k
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
15
votes
1
answer
2k
views
Good introduction to Morse-Novikov theory?
Morse theory investigates the topology of compact manifolds using critical points of real-valued functions $f\colon\, M\to \mathbb{R}$. Motivated by problems in dynamical systems, Novikov (Multivalued ...
- 22.1k
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
4
votes
1
answer
434
views
Curvature and Symmetry on Kähler manifolds
Hi there,
Suppose $X$ is a Kähler manifold that has an analytic isometry $S$, with $S^k = \operatorname{Id}$ ($k \in \Bbb N$). In a situation like this (maybe with additional assumptions on $X$) can ...
- 1,211
7
votes
1
answer
865
views
Associated vector bundles of infinite rank and induced connections
Let $\mathbb{V}$ be a representation of a Lie group $G$ and let $P \to M$ be a principal $G$-bundle with a principal connection. If $\mathbb{V}$ is finite-dimensional, then one can associate to this ...
- 8,597
1
vote
0
answers
221
views
Co-normal bundle of orthogonal compliment
Is the following fact well known?
Let $X$ be a manifold and $V$ be a vector space. Let $E_1$ be a sub-bundle of the constant bundle $X \times V$. Let $E_2$ be its orthogonal compliment in $X \...
- 2,649
6
votes
1
answer
723
views
Geometric treatment of the Ward-Takahashi identity
The quantum field theory generalisation of Noether's theorem about symmetries and conservation laws is the Ward-Takahashi identity.
What is a suitable treatment of this in the context of differential ...
- 921
17
votes
3
answers
2k
views
Laplacians on graphs vs. Laplacians on Riemannian manifolds: $\lambda_2$?
A graph $G$ is connected if and only if
the second-largest eigenvalue $\lambda_2$ of
the Laplacian of $G$ is greater than zero.
(See, e.g.,
the Wikipedia article on algebraic connectivity.)
Is ...
- 151k
2
votes
1
answer
308
views
Connecting tangents of convex curves: at some point orthogonal?
Let $a(t)$ and $b(t)$ be two smooth, nested convex curves in the plane, $t\in[0,1]$:
Suppose the parametrization of $a()$ and $b()$ is such that $\dot{a}(t)$ is ...
- 151k
5
votes
0
answers
350
views
Areas dominated by two points on a surface: Equal?
Let $S$ be a smooth compact surface in $\mathbb{R}^3$, with two distinct, distinguished points
$a,b \in S$. Let $R(a)$ be all the points of $S$ closer to $a$ than to $b$, and $R(b)$ all the
points of ...
- 151k
14
votes
5
answers
4k
views
References for classical Yang-Mills theory
I am looking for a reference to study classical (i.e., not quantized) Yang-Mills theory.
Most of the sources I find focus on mathematical aspects of the theory, like Bleecker's book Gauge theory and ...
- 1,064
15
votes
1
answer
1k
views
Lie algebra valued 1-forms and pointed maps to homogeneous spaces
Let $G$ be a Lie group with Lie algebra $\mathfrak{g}$, and let $(M,p_0)$ be a simply connected pointed smooth manifold. A $\mathfrak{g}$-valued 1-form $\omega$ on $M$ can be seen as a connection form ...
- 6,777
1
vote
0
answers
362
views
Archimedes’ and Galileo’s spirals in one equation.
The differential equation in polar coordinates
$r'^2+r^2=(kt)^2$, $r(t=0)=0$, k- Const, for large $t$ presents Archimedes’ Spiral and Galileo's spiral for $t \to 0$.
I find it surprisingly, however ...
28
votes
2
answers
3k
views
Probing a manifold with geodesics
Supposed you stand at a point $p \in M$ on a smooth 2-manifold $M$
embedded in $\mathbb{R}^3$.
You do not know anything about $M$.
You shoot off a geodesic $\gamma$ in some direction $u$,
and learn ...
- 151k
2
votes
2
answers
411
views
epsilon-Manifold with curvature at one point
I remember briefly hearing about this notion (stated in the title), of a manifold where there is a nonzero curvature at precisely one point (a delta-function distribution), and such that there is a ...
- 17.5k
8
votes
1
answer
1k
views
Calculating a curvature tensor by polarization
I'm reading some articles by Siu and Nannicini on the Weil-Petersson metric associated to families of compact Kahler-Einstein manifolds. In each article Siu and Nannicini construct a Weil-Petersson ...
- 4,343
26
votes
2
answers
2k
views
Why is the half-torus rigid?
The half-torus surface that results from slicing a torus like a bagel,
depicted below (left), is isometrically rigid.
I know this from a remark of Alexandrov in
Mathematics: Its ...
- 151k
36
votes
10
answers
6k
views
Determining a surface in $\mathbb{R}^3$ by its Gaussian curvature
A curve in the plane is determined, up to orientation-preserving
Euclidean
motions, by its curvature function, $\kappa(s)$.
Here is one of my favorite examples, from
Alfred Gray's book,
Modern ...
- 151k
34
votes
1
answer
6k
views
Jet bundles and partial differential operators
A geometric way of looking at differential equations
In the literature for the h-principle (for example Gromov's Partial differential relations or Eliashberg and Mishachev's Introduction to the h-...
- 39k
6
votes
5
answers
3k
views
Navier-Stokes equations in Riemannian geometry
The Navier-Stokes equations can be written on a Riemannian manifold as:
$$\dot{u}+\nabla_u u+ \Delta u=(df)^* $$
$$d^* u=0$$
where $\nabla$ is the Levi-Civita connection, $u$ is a vector field, $\...
- 663
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
15
votes
4
answers
1k
views
Geodesics in $\mathbb{R}^2 \times \mathbb{S}^1$ under "segment" metric
Represent the position of a unit-length, oriented segment $s$ in the plane
by the location $a$ of its basepoint and
an orientation $\theta$: $s = (a,\theta)$. So $s$ can be
viewed as a point in $\...
- 151k
8
votes
3
answers
1k
views
Higher derivatives than Jacobi fields
The first and second derivatives of the distance function (either the full $d:M\times M\to \mathbb{R}$ function or the $d(p,\cdot):M\to \mathbb{R}$ function) as well as the derivative of the ...
- 516
7
votes
1
answer
1k
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Helmholtz-Decomposition on compact Riemannian manifolds
For smooth domains $\Omega$ in $\mathbb{R}^n$ it is known that one can decompose vector fields in $L^p(\Omega)^n$, $1 < p <\infty $ into a "gradient"- and a "divergence-free"-part such that
$L^...
- 73
7
votes
2
answers
1k
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A book on Banach Manifold for a Dynamicist
Hi all,
Could you give me a suggestion of suitable book about Banach Manifolds for someone that have background in functional analysis at the level of Conway's book and Do Carmo's book on Riemannian ...
user11178
5
votes
4
answers
2k
views
Relationship between the focal locus and the cut locus
I am seeking
clarification of
the relationship between the
focal locus
and the
cut locus
of a curve $C$ in $\mathbb{R}^2$, and
of a surface $S$ in $\mathbb{R}^3$.
Essentially my question is,
Under ...
- 151k
2
votes
0
answers
115
views
Special class of bi-hamiltonian systems
A bi-Hamiltonian manifold is a manifold $M$ equipped with two compatible Poisson tensors $\pi_0$ and $\pi$.
I am interested in the case of a Lie group $G$ endowed with a multiplicatif Poisson tensor $...
- 513
22
votes
11
answers
9k
views
Maxwell's equations and differential forms
Is there a textbook that explains Maxwell's equations in differential forms?
What I understood so far is that the $E$ and $B$ fields can be assembled to
a 2-form $F$, and Maxwell's equations can be ...
7
votes
6
answers
1k
views
Developable 3-manifolds in $\mathbb{R}^4$
Is there a classification of the equivalent of a "developable surface" in $\mathbb{R}^4$?
Analogous to: planes, cylinders, cones, and tangent developables in $\mathbb{R}^3$?
Edit: Here I am imagining "...
- 151k
5
votes
2
answers
1k
views
On the smooth structure of the spaces of $k$-jets
I was asking myself, if the following list of conditions is sufficient to determine the usual smooth structure on the spaces of $k$-jets.
the map $j^k f:M\ni x\to j_x^k f\in J^k(M,N)$ is smooth, for ...
- 4,306
7
votes
2
answers
787
views
Shortest paths on linked tori
I will make this question specific at first, and general later.
Suppose we have two linked tori, $T_1$ and $T_2$,
each of radii $(2,1)$, meaning that each torus is the result of sweeping
a circle of ...
- 151k
2
votes
1
answer
347
views
Lee codes and $n$-torus
This is in continuation with this post:
Geometric/Analytic techniques for constructive and asymptotic bounds in the Lee metric
Codes over alphabet $\mathbb{Z}_{q}$ of length $n$ for the Lee metric ...
- 800
6
votes
0
answers
437
views
Has anyone seen this Hitchin-like system?
Let $(M,g)$ be a riemannian manifold and let $P\to M$ be a principal $G$-bundle with connection $A$. Let $\alpha \in \Omega^1(M;\mathrm{ad}P)$ be a one-form on $M$ with values in the adjoint bundle $\...
- 33.3k
8
votes
1
answer
787
views
The rain hull and the rain ridge
Rain falls steadily on an island, a 2-manifold $M$, which you may
assume, as you prefer,
is: (a) smooth, or (b) a PL-manifold, or perhaps even
(c) a
triangulated irregular network (TIN).
After a time,...
- 151k
8
votes
2
answers
2k
views
Estimates on the Green function of an elliptic second order differential operator.
Let $D$ be a linear differential elliptic operator of second order
with infinitely smooth coefficients acting on real valued functions
on a compact manifold $M$. Let us assume that $D$ has no free ...
- 21.8k