All Questions
Tagged with reference-request dg.differential-geometry
800 questions
9
votes
1
answer
456
views
Minimal immersions of the 2-sphere
Following the ideas of S.-S. Chern, J. L. M. Barbosa associated a holomorphic curve in $\mathbb{C}P^m$ to a minimal immersion of the 2-sphere into the $2m$-sphere in his 1972 paper. However, his ...
- 427
2
votes
1
answer
164
views
Curvature computations of globally symmetric spaces of rank $1$
I've been having trouble with finding the curvature computations of globally symmetric spaces of rank $1$.
More specifically, I need to use results about the eigenvalues of the operator $R:T_pM \...
- 401
18
votes
2
answers
1k
views
formula for Eta invariant
Hirzebruch's signature formula is not valid for manifolds with boundary.
An error term is introduced by Atiyah-Patodi-Singer to fix it.More precisely:
$$sign (M)=L(M)[M]+\eta(\partial M)$$
Yet ...
- 259
5
votes
3
answers
550
views
Can the conformal structure on the projective light-cone detect hyperplane sections?
Let $(V,\langle\,\cdot\,,\,\cdot\,\rangle)$ be an $(n+1)$-dimensional real vector space, equipped with a nondegenerate symmetric bilinear form of indefinite signature, and denote by $\nu(v):=\langle v,...
- 2,527
8
votes
1
answer
421
views
$C^k$ one-parameter family of metrics
Consider a smooth Riemannian manifold $M$ and a $C^k$ one-parameter family of Riemannian metrics $g_t$ on $M$. Here $k$ could be any integer, $k$ could be infinity, when the one-parameter family $g_t$ ...
- 1,407
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
0
votes
1
answer
291
views
Where is the paper "Theorie de Lie pour les groupoides differentielles (J. Pradines)"?
Can anyone help me finding the paper:
"Theorie de Lie pour les groupoides differentielles (J. Pradines)"
I'm researching Lie groupoids and I was refered to that paper several times but couldn't find ...
- 1
4
votes
3
answers
5k
views
Green's function on sphere
Consider radial (normal) coordinates on a sphere $S^n, n \geq 2$. Let the "origin" be the north pole $(0, 0,..., 1)$ and the coordinates be denoted by $(r, \theta)$. We know that the Laplacian $\...
- 41
5
votes
0
answers
1k
views
Prerequisites for reading Gregory Perelman's work
What are the prerequisites for understanding the work of Perelman concerning the Poincaré conjecture?
I am referring to the last three papers here.
- 1,594
4
votes
0
answers
293
views
Cohomology of the classifying space of some Super Lie group
Are there any papers on the cohomology of the classifying space of the general linear supergroup $GL(n, m)$ or unitary supergroup $U(n, m)$?
I know basically nothing about supergeometry. It seems ...
- 1,173
2
votes
1
answer
83
views
Calculating the Upper Bound on the Sphere Radius of Knotted Channel Surfaces
This question is motivated by trying to determine the upper bound on the thickness of a rope of fixed length (w.l.o.g. $2\pi$), with which a knot of given topology can be realized under the further ...
- 13.2k
4
votes
2
answers
758
views
Riemannian metric of hyperbolic plane
I'm fishing for the origin of the idea to consider "trace scalar product" on the space of ($G$-)orthogonal projectors as means of defining a Riemannian metric on some subset of lines in a vector space....
- 8,597
4
votes
2
answers
505
views
Eigenfunctions of the Laplacian on singular spaces
Consider a compact manifold $M$ with boundary and corner. As an example, we could have the cube $\{(x_1, x_2,..x_n) \in \mathbb{R}^n : x_i \in [0,1]\}$. We could very well define the Laplacian $\Delta$...
- 43
35
votes
8
answers
19k
views
Modern mathematical books on general relativity
I am looking for a mathematical precise introductory book on general relativity. Such a reference request has already been posted in the physics stackexchange here. However, I'm not sure whether some ...
- 1,550
2
votes
0
answers
58
views
Mathematical Billiards: Set of starting points and velocities hitting boundary at time t
In the simplest setting $\Omega$ smooth compact and convex in $R^n$ with linear constant speed trajectories that is ($q_t=q_0+t\cdot v$ until the collision point). What is known about the structure ...
- 1,256
48
votes
2
answers
14k
views
Research situation in the field of Information Geometry
I am now doing an article survey on the field of information geometry started by S.Amari and Barndorff-Nielson. I want to know some research situation in this field.
I have read (4) and parts of (3). ...
- 8,071
64
votes
12
answers
22k
views
Advanced Differential Geometry Textbook
I tried this post on StackExchange with no luck. Hopefully the experts at MathOverflow can help.
In algebraic topology there are two canonical "advanced" textbooks that go quite far beyond the usual ...
4
votes
0
answers
301
views
Was this particular case of the tube formula known before Weyl and Hotelling?
The tube formula is a really nice result in differential geometry which relates the volume of the tubular neighborhood of a submanifold to its intrinsic geometry. It has been proved by Weyl in 1939 ...
- 4,123
17
votes
2
answers
3k
views
Open questions in "Spin geometry"
This is a very naive question. I have the impression that the area of "Spin geometry" is not an active research field. Sure Spin geometry is used in many different branches of mathematics and physics ...
- 2,816
2
votes
0
answers
59
views
Group of real analytic isometries of $g$-fold product of the Poincare upper half plane
Let $\mathfrak{h}^g$ be the cartesian product of $g$ copies of the Poincare upper half plane. We endow $\mathfrak{h}^g$ with the usual Poincare metric given in local coordinates by $ds^2=\sum_{i=1}^g ...
- 7,609
8
votes
1
answer
1k
views
Example of a triangulable topological manifold which does not admit a PL structure
I know there are some examples of manifolds which don't admit a PL structure (combinatorial triangulation), and that it has been recently proven that in dimension $n\geq5$ there are manifold which are ...
- 683
60
votes
1
answer
6k
views
What were the main ideas and gaps in Yoichi Miyaoka's attempted proof (1988) of Fermat's Last Theorem?
Out of sheer curiosity I have been reading Stewert and Tall's "Algebraic Number Theory and Fermat's Last Theorem" (2001). As it contains various bits of history, I found out to my own shame that I was ...
- 7,127
8
votes
3
answers
2k
views
What does it mean that the Hessian is proportional to the metric?
Let $(M,g)$ be a smooth manifold equipped with a metric tensor $g$, and $f\in C^\infty(M)$ a regular function (i.e., with nowhere vanishing differential).
Denote by $\mathrm{Hess}_g(f):=\nabla df$ ...
- 2,527
6
votes
0
answers
493
views
Reference for the Banach Manifold structure of $C^k(M,N)$
I'm completely new to the subject of banach manifolds and I'm looking for a reference of the following:
Let $M$,$N$ be smooth (=$C^\infty$) finite-dimensional compact manifolds. Consider the set $C^...
- 71
6
votes
1
answer
185
views
Zeta-Determinant for shifted Laplacians on the circle
Consider on the circle $S^1$ the operator
$$L := - \frac{\partial^2}{\partial \theta^2} + c$$
for some constant $c \in \mathbb{R}$.
What is its $\zeta$-regularized determinant?
This should be well-...
- 9,853
8
votes
2
answers
1k
views
Survey papers on the role played by PDE in mathematics
There are already several questions on MathOverflow that inquire about the many diverse relationships between PDE and several other 'areas' of mathematics (e.g., algebraic and differential geometry ...
7
votes
0
answers
407
views
Linear vs smooth actions of finite groups on spheres, euclidean spaces and closed disks
I would like to know examples (with references, if possible) of the following:
(1) a finite group $G$ acting effectively and smoothly on a sphere $S^n$ (any $n$) but admitting no effective linear ...
10
votes
2
answers
484
views
References on quaternionic geometry
Is there any analog, in the quaternionic setting, of Kahler potentials?
In particular I'm interested in a similar construction of the Fubini-Study 2-form on $\mathbb{P}^1(\mathbb{C})$ over the ...
- 101
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
0
votes
1
answer
226
views
Marcel Berger's "Sur les groupes d'holonomie homogènes de variétés à connexion affine et des variétés riemanniennes."
I would appreciate any reference that contains either a translation or proof of the main theorem in this paper. Thank you in advanced.
2
votes
1
answer
333
views
Spherical harmonics and ellipticity of the Laplacian
Let us consider the sphere $S^n$ and the Laplacian $-\Delta$ on it. Let $L^2(S^n) = \bigoplus_k V_k$, where $V_k$ represents the eigenspace of the Laplacian with eigenvalue $k(k + n - 1)$. We know ...
- 1,407
4
votes
1
answer
733
views
Reference request: Intrinsic definition of the strong Whitney topology on $\mathcal{C}^{\infty}(M,\mathbb{R})$ without using charts or jets
Let $M$ and $N$ be smooth manifolds. There is a description of the strong Whitney topology on $\mathcal{C}^{\infty}(M,N)$ in terms of partial derivative in charts (using locally finite sets of charts ...
- 191
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
5
votes
1
answer
198
views
Generalisation of "tangent space" to not-necessarily connected sets
I vaguely recall having read somewhere a definition similar to (but probably not exactly the same as) the following.
Definition (Blob) Let $S\subset \mathbb{R}^n$ be a set, and $p \in S$. The Blob ...
- 39k
1
vote
1
answer
732
views
Norm equivalent to Sobolev norm? [closed]
On the hyperbolic space $\mathbb{H}^n$, it is known that the spectrum of the Laplacian satisfies $\text{Spec}(-\Delta) \subset [\frac{(n - 1)^2}{4}, \infty)$. Consider the operator $P = -\Delta + a$, ...
- 11
0
votes
0
answers
234
views
What is the symplectic manifold whose Delzant polytope is a trapezoid?
What is the symplectic form on the manifold whose associated Delzant polytope is a trapezoid? I am trying to find it by using the Marsden–Weinstein theorem, but I have been unable to do so. If ...
- 1
4
votes
0
answers
111
views
Integrability of Continuous Tangent Subbundles
Are there any field of mathematics, except dynamical systems, where one needs to integrate continuous sub-bundles of the tangent space?
More specifically given a smooth manifold of $M$ and a ...
- 419
18
votes
2
answers
4k
views
Where is the exponential map a diffeomorphism?
Let $M$ be a closed compact Riemannian manifold.
The exponential map $\mathrm{exp}:TM\to M\times M$ takes $(p,v)$ to $(p,\gamma_v(1))$, where $\gamma_v$ is the geodesic flow at $p$ in the direction ...
- 4,588
12
votes
1
answer
2k
views
Sard's Theorem For Banach Spaces
Given a smooth map from $\phi: B \rightarrow M$ where $B$ is a Banach Space and $M$ is a finite dimensional smooth manifold (for example, the end point map for a control system), what is the strongest ...
- 2,099
17
votes
2
answers
2k
views
Differential operators are coKleisli morphisms of the jet co-monad
The following statement may be "well known but not well known enough", and my question is which reference would state it explicitly:
The construction of Jet bundles is a comonad on suitable bundles ...
- 19.8k
1
vote
0
answers
96
views
Largest dimensional Lie subgroup of $SU(N)$ [duplicate]
What is the largest (Lie) subgroup of $SU(n)$ in the sense of its dimension.
I am aware of this potential duplicate subgroup of SU(N) with maximal manifold dimension , however, the title of this ...
- 2,099
16
votes
4
answers
2k
views
Inverse problem of Chern Classes
For my graduate (master) thesis I am studying the theory of Chern Classes. As a possible personal development the only sensible idea I have so far, and which I frankly think is impossible, is to work ...
- 363
8
votes
1
answer
911
views
Avoiding mean-curvature flow dumbbell neck-pinch by inflating a surface
It is well known that
Grayson's dumbbell neck-pinch1,2 separates
into disconnected pieces under
mean curvature flow:
Image ...
- 151k
2
votes
0
answers
108
views
Quantitative estimate of heat dispersion - off diagonal estimates
Consider the heat equation $\partial_t u - \Delta u = 0$ on a compact manifold $M$ (if $M$ has a smooth boundary, then we assume either Dirichlet or Neumann boundary condition). Consider $u_0 (x) = u(...
- 21
5
votes
3
answers
1k
views
Constant rank theorem for Banach spaces
Is there a similar statement to the constant rank theorem for finite dimensional real smooth manifolds which holds for a smooth map $F:B \rightarrow M$ where $B$ is an infinite (countable) dimensional ...
- 2,099
7
votes
2
answers
1k
views
References for the moduli space of complex structures
I am looking for references where the moduli space of complex structures on a complex manifold is well explained: in particular the infinitesimal deformations, the obstructions, the elliptic complex ...
- 2,816
2
votes
0
answers
168
views
Sources on evolution of submanifolds subject to Ricci flow
I am seeking any textbook or paper addressing the evolution of submanifolds of a manifold undergoing Ricci Flow. Please, any pointer towards this topic is more than welcome.
This old MO post may be ...
- 153
1
vote
1
answer
252
views
Sobolev multiplication $\otimes$ of $H^1=W^{1,2}$ in vector bundles
Let $E\to X$ be a vector bundle with an inner product and fix a reference connection $A_0$ on $E$. Then for $1\leq p < \infty$ and $k\geq 0$ we can define the Sobolev space $W^{k,p}(E)$ as the ...
- 729
11
votes
1
answer
2k
views
A survey for various $K$-homology theories and their relationship
The ordinary Topological $K$ theory defined by Atiyah and Hirzebruch is a generalized cohomology theory (see wikipedia).There is the Bott spectrum associated to this generalized cohomology theory....
- 593
23
votes
3
answers
2k
views
Is the analytic version of the Whitney Approximation Theorem true?
I initially asked this question on MSE but I haven't had any luck.
The Whitney Approximation Theorem states that any continuous map between smooth manifolds is homotopic to a smooth map. If the ...
- 19.3k