All Questions
Tagged with reference-request dg.differential-geometry
800 questions
10
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3
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541
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Curvature of the boundary vs. normal derivative of the first eigenfunction
Disclaimer. I posted this question in Math.SE, but it haven't received enough attention.
Let $\varphi_1$ be the first eigenfunction of the zero Dirichlet Laplacian in a planar bounded domain $\Omega$....
- 201
10
votes
0
answers
426
views
Gromov's compactness theorem via Sacks-Uhlenbeck and Schoen-Uhlenbeck
Gromov's compactness theorem for pseudo-holomorphic curves, in section 1.5 of "Pseudoholomorphic curves in symplectic manifolds," is very well known. I'm aware of the following two proofs:
...
- 2,247
10
votes
0
answers
216
views
Can we find minimal-diameter metrics without computability?
A beautiful argument by Nabutovsky and Weinberger (see http://math.uchicago.edu/~shmuel/fractal.ps) shows that, if $M$ is any smooth compact manifold of dimension $\ge 5$, then the diameter functional ...
- 21.2k
10
votes
0
answers
284
views
Comparing spectra of Laplacian and Schrödinger operator
Let $M$ be a closed (compact without boundary) Riemannian manifold. Is there a body of results that compares the eigenvalues of the Laplace-Beltrami operator with that of Schrödinger operators $-\...
- 109
10
votes
0
answers
415
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Singularities in Yang Mills Flow
In "The Yang Mills flow in four dimensions", M. Struwe proves that this flow converges, up to bubbling phenomena. And he has conjectured that this explosion in finite time should happen as proven for ...
- 914
9
votes
3
answers
2k
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Ricci flow with surgery in dimension 2
Is it possible to define the Ricci flow with surgery in dimension 2 and use it to classify the surfaces?
I know this is overkill, there are simpler ways to classify surfaces, but I would like to ...
- 3,033
9
votes
6
answers
4k
views
Books for hyperbolic geometry ( surfaces ) with exercises?
what are good books on hyperbolic geometry/hyperbolic surfaces that have good number of exercises, just to get a good understanding of the literature . I know John Ratcliffe's book will be one of them,...
- 1,471
9
votes
5
answers
1k
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List of generic properties of Riemannian metrics
I am highly interested in compiling a list of generic properties of Riemannian metrics on a (may be compact) manifold in general, or under "relatively broad" assumptions, like generic properties of ...
9
votes
1
answer
874
views
Proofs that the conformal group in dimension $\ge 3$ is a Lie group
Let $M$ be a smooth manifold of dimension $\ge 3$, equipped with a conformal structure (or a Riemannian metric). Then, the group of conformal diffeomorphisms is a finite dimensional Lie group.
A ...
- 6,741
9
votes
5
answers
2k
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A survey on positive mass theorem?
Could you suggest a good survey paper on positive mass theorem?
9
votes
1
answer
2k
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Is a manifold generically real analytic (with generic real analytic metric)?
I have heard it said in some differential geometry talks that "the generic situation in such and such case is real analytic". My question is, is the generic smooth manifold also real analytic in some ...
- 123
9
votes
2
answers
981
views
Text on old-fashioned differential geometry
I am seeking good books on the geometry of surfaces in Euclidean space, which would in particular discuss Darboux frames. Please explain for each suggestion why you like this book (classics are ...
- 14.4k
9
votes
2
answers
2k
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Surfaces in $\mathbb R^3$ with negative curvature bounded away from zero
Is there a surface in $\mathbb R^3$ which is a closed subset and whose curvature is negative and bounded away from zero?
And the small-print...
By surface I mean smooth surface without boundary, and ...
- 47.3k
9
votes
2
answers
7k
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Constant curvature manifolds
In two different books I found these two related statements.
The book by Jost defines a ``locally symmetric space" as one for which the curvature tensor is constant and which is geodesically complete....
- 3,541
9
votes
1
answer
2k
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Is there a book on differential geometry that doesn't mention the notion of charts?
What are some books/texts that use chart free coordinate free language for things otherwise written in a coordinate based formulation? I would like to learn about covariant differentiation, curvature, ...
- 319
9
votes
2
answers
267
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group actions in dimension 2 and 3
I am looking for a reference to the following claims:
Any compact group (connected or not) acting on $S^2$ is differentiably conjugate to a linear action. This must be classical.
A circle $S^1$ ...
9
votes
2
answers
639
views
Contact distributions on $(G_2,P)$-type Cartan geometries in dimension 5
Up to topology, the 5D homogeneous space
$$
G_2/P
$$
of the (real form of the) 14D exceptional Lie group $G_2$ is the 5D jet space
$$
M:=J^1(2,1)=\{(x,y,u,p,q)\}
$$
of scalar functions in two ...
- 2,527
9
votes
2
answers
925
views
differential geometry using Robinson's infinitesimals?
Is there a detailed treatment of differential geometry using Robinson's infinitesimals?
- 16.6k
9
votes
1
answer
545
views
Citation hunting: Floer on spectral sequences
I vaguely remember a YouTube talk that began with a citation from Floer regarding the existence of a spectral sequence. The idea was that given a manifold with a Morse function, we can construct a ...
- 193
9
votes
4
answers
469
views
Notion of smoothness for set-valued functions
Is there a way of talking about continuity and smoothness for set valued functions? More precisely, consider $M$ and $N$ topological/smooth manifolds, and let $f$ a function that associates to each ...
- 39k
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
9
votes
1
answer
3k
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Oloid and sphericon: rolling develops entire surface
Wikipedia says that,
"The oloid is one of the only known objects, along with some members of the sphericon family, that while rolling, develops its entire surface."
Below are illustrations of ...
- 151k
9
votes
1
answer
4k
views
Complex geometry text/research introduction for the analyst
To give some background, I am mainly an analyst trained in harmonic/functional and do work on geometric pde's and spectral multipliers. Of late, I am trying to learn more about (research level) ...
- 141
9
votes
1
answer
509
views
A question on generalized Einstein metrics on four-dimensional manifolds
I am thinking of a possible generalization of Einstein metrics (or a possible characterization of Einstein metrics) on four-dimensional manifolds,
\begin{equation*}
\mathrm{Ric}\circ\mathrm{Ric}=\...
- 399
9
votes
2
answers
399
views
Reference for a path groupoid being a diffeological groupoid
I am looking for a reference that has a proof that a path groupoid
is a groupoid internal to the category of diffeological spaces. I do know how to prove this fact, and a proof is not hard. My reason ...
- 1,710
9
votes
2
answers
478
views
Topology of the Universal Spinor Field Bundle
While reading article [1] below I came across the notion of a universal spinor bundle. This is defined at the beginning of section 6 (p.14) in [1] as follows: Let $M$ be a spin manifold and $\mathcal{...
- 408
9
votes
1
answer
630
views
$C^{k,\alpha}$ diffeomorphisms and vector fields
This feels like something I should know, but I have a hard time finding a definite reference.
Let $M$ be a compact (Riemannian) manifold, $k\ge 1$ be an integer and $\alpha\in(0,1)$. When v is a $C^k$...
- 14.4k
9
votes
1
answer
581
views
Smoothing of a Kähler orbifold metric on a complex surface
Let $S$ be a smooth complex projective surface and $D\subset S$ be a smooth complex curve. Fix an integer $m>1$ and consider $(S,D,m)$ as an orbifold with orbi-locus $D$ with stabilizer $\mathbb ...
- 14.3k
9
votes
0
answers
996
views
Complexification of smooth manifolds
Given a $C^\infty$- smooth manifold $M$, is there a natural way to complexify it?
By this I mean finding a complex manifold $N$ and a smooth function $f:M\to N$ such that $df:T_xM\otimes \mathbb{C}\to ...
- 435
8
votes
2
answers
1k
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Is there an English translation of Minding's 1839 paper?
Is there an English translation of "Wie sich entscheiden lässt, ob zwei gegebene
krumme Flächen auf einander abwickelbar sind oder nicht..."
by Ferdinand Minding, Journal für die reine und angewandte
...
- 305
8
votes
2
answers
2k
views
Differential forms, PDE's and Élie Cartan
Hello everybody, I would like to know about the work of Élie Cartan of PDE's that relate to the theory of foliations and differential forms.
I am interested in the subject and will be happy to ...
- 83
8
votes
4
answers
710
views
Torsion of submanifolds
Studying curves in the Euclidean three dimensional space, one usually defines the curvature and the torsion of a curve. If I am not missunderstanding the thing, I guess that a curve has zero torision ...
- 2,384
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
8
votes
3
answers
976
views
Examples and properties of spaces with only trivial vector bundles
Let $B$ be a paracompact space with the property that any (topological) vector bundle $E \to B$ is trivial. What are some non-trivial examples of such spaces, and are there any interesting properties ...
- 1,763
8
votes
3
answers
914
views
Spectral sequences in algebraic topology [duplicate]
What books/articles do you recommend for learning spectral sequences? I am interested in their applications to algebraic topology, particularly to understand the homology of fibre bundles. I have a ...
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
8
votes
2
answers
350
views
Compressible Ebin-Marsden?
In Ebin and Marsden's paper Groups of Diffeomorphisms and the Motion of an Incompressible Fluid, there is a footnote on the first page indicating that non-homogeneous cases and the case of ...
- 39k
8
votes
1
answer
856
views
Are there mistakes in Kovalev's "Twisted connected sums and special Riemannian holonomy"?
This is kind of a strange and vague question... sorry about that.
I am really interested in $G_2$ Twisted Connected sums as described in this paper: https://arxiv.org/abs/math/0012189 "Twisted ...
user117832
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
8
votes
1
answer
2k
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K.Uhlenbeck's preprint "A priori estimates for Yang-Mills fields"
Does anyone have a copy of the unpublished preprint of Karen Uhlenbeck A priori estimates for Yang-Mills fields from around 1986?
It appears to have circulated for some time, and it is quoted in ...
- 6,871
8
votes
1
answer
476
views
Closed geodesics on constant positive Gauss curvature surfaces
Can we show that geodesics with a rational radius $ r_{mid-equator} =a q/p \,(p>q ) $ at mid-equator on a spindle type surface of revolution of constant Gauss curvature $ (K=1/a^2 \, $in $\, \...
- 917
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2
answers
1k
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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 ...
8
votes
2
answers
827
views
Any text book or lecture notes regarding the algebraic part of geometry?
I know there are text books of Algebraic topology. There are books of Differential geometry. But when I read papers, for example lots of papers talking about fundamental groups or higher homotopy ...
- 2,623
8
votes
1
answer
1k
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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
8
votes
1
answer
1k
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Spectrum of the Laplacian on p-forms on the sphere
In this paper the authors give an explicit description of the eigenforms and spectrum of the Laplacian acting on $p$-forms on the round sphere $S^n$, apparently citing an unpublished computation of ...
- 585
8
votes
1
answer
633
views
Flag manifolds (=R-spaces): quotients by parabolic subgroups vs. isotropy representation
Real flag manifolds (also known as R-spaces) can be defined in two ways which I believe are equivalent although some fine print may have escaped me:
as a quotient of a semisimple real Lie group $G$ ...
- 32.5k
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
8
votes
1
answer
306
views
Fibers of generic smooth maps between manifolds of equal dimension
I have heard that the following is a "well-known"
Claim. Let $M$ and $N$ be smooth manifolds with equal dimensions and $M$ compact. Then a generic smooth map $f\colon M\to N$ has finite ...
- 501
8
votes
1
answer
682
views
Geometry of convex sets in Riemannian manifolds
Let $M$ be a smooth Riemannian manifold without boundary. Let $X\subset M$ be a closed subset which is a smooth submanifold with boundary, $\dim X=\dim M$. Assume that $X$ is locally convex, i.e. any ...
- 21.8k
8
votes
2
answers
377
views
Surfaces contained in a ball
In this Paper there is a proof that a closed plane curve of length
$L$ and curvature bounded by $K$ can be contained inside a circle of radius
$L/4 - (\pi - 2)/2K$. Are there similar results for ...
- 1,252