All Questions
Tagged with reference-request dg.differential-geometry
800 questions
7
votes
4
answers
2k
views
$E$ is a holomorphic vector bundle if and only if there is a Dolbeault operator $\bar{\partial}_E$
I am looking for a reference which shows that the following statements are equivalent for a complex vector bundle $E$:
$E$ is a holomorphic vector bundle.
There is a Dolbeault operator $\bar{\partial}...
- 19.3k
1
vote
2
answers
205
views
Tangent vectors on the algebra of trigonometric polynomials
Let $G$ be a compact real Lie group and ${\sf Trig}(G)$ the algebra of trigonometric polynomials on $G$ (defined in the Hewitt-Ross, Abstract harmonic analysis, (27.7)), i.e. the algebra of functions $...
- 7,426
4
votes
2
answers
375
views
Converse to Chow's theorem in sub-riemannian geometry
Chow's theorem is the statement that if $M$ is a connected smooth manifold endowed with a distribution $\mathcal{D}$ which is completely non integrable (i.e. iterated commutators of smooth sections of ...
- 2,479
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
17
votes
2
answers
2k
views
Where did Sophus Lie write the group commutator for two one parameter groups
If $X,Y$ are vector fields and $\def\Fl{\operatorname{Fl}}\Fl^X_t$ and $\Fl^Y_t$ their local flows, let $[\Fl^X_t,\Fl^Y_t]:= \Fl^Y_{-t}\Fl^X_{-t}\Fl^Y_t\Fl^X_t$ denote the group commutator of the ...
- 25.3k
6
votes
2
answers
339
views
Continuity of the spectrum with respect to the metric
The following question is quite natural, but I am not aware of a reference dealing with it: let $M$ be a compact (smooth) manifold (posssibly with boundary) and $E$ a vector bundle on $M$ with an ...
- 3,399
34
votes
1
answer
4k
views
Strong Whitney embedding theorem for non-compact manifolds
$\newcommand{\RR}{\mathbb{R}}$The present question arises from some confusion on my part regarding the precise statement of the strong Whitney embedding theorem for non-compact manifolds.
The strong ...
- 6,210
7
votes
1
answer
554
views
Lower bound on $L^2$ norm of mean curvature in general dimensions
Suppose $\Sigma\subset \mathbb{R}^{n+1}$ is a closed embedded hypersurface. We know that when $n=1$
$$
\int_{\Sigma} |H|^2 \geq \frac{4 \pi^2}{|\Sigma|}
$$
by Gauss-Bonnet and that this is saturated ...
- 2,299
1
vote
1
answer
574
views
Proof that the Hodge-de Rham Rank Equals the Euler Characteristic
Can someone please provide a good (online accessible) reference for the well-known identity
$$
\text{rank((d + d}^*)^+) = \sum_{i=}^n (-1)^i \dim(H^i(M)),
$$
where $M$ is a manifold of dimension $n$, ...
- 11
26
votes
2
answers
1k
views
Vector fields on $(4n+1)$-spheres
If $n$ is odd then $S^{n-1}$ doesn't admit a nowhere-vanishing vector field, and if $n$ is even then there does exist one (Hairy Ball Theorem). We can then ask, on $S^{n-1}$, what is the maximum ...
- 17.5k
6
votes
3
answers
590
views
When does one obtain different 3-manifolds by pasting two tori?
Consider a compact solid torus $T$ and a diffeomorphic copy of it $T' \subset T$ embedded in the interior of $T$ in such a way that it makes two turns around the central circle of $T$.
I would like ...
- 4,304
6
votes
2
answers
1k
views
Parallel forms and cohomology of symmetric spaces
Let $G/H$ be a compact symmetric space. Then I believe the following is true: if $\alpha \in \Omega^k(G/H)$ and $\nabla$ the Levi-Civita connection, then
$$
(\alpha \text{ is induced by an $\...
- 3,244
9
votes
2
answers
927
views
differential geometry using Robinson's infinitesimals?
Is there a detailed treatment of differential geometry using Robinson's infinitesimals?
- 16.6k
17
votes
3
answers
1k
views
On closed simple curve with curvature at most 1
I am looking for the reference to the following theorem.
I have to apply a similar statement, and it would be nice to trace the source.
Please note, I know few proofs in fact it is Problem 3 in my ...
- 45k
1
vote
0
answers
119
views
Orbits and indices of vector fields
I'm afraid this might be an exercise in differential topology (in which case a reference to a book where it is would be very much appreciated); apologies in advance. Given an analytic vector field (in ...
- 4,432
19
votes
3
answers
2k
views
what is a spinor structure?
There are of course lots of definitions and references for this, but in the same way that, on a manifold $M$,
a Riemannian metric is a section of positive definite symmetric bilinear forms on $TM$
or ...
- 4,432
1
vote
0
answers
382
views
Question in the paper of Robert Bryant "Calibrated embeddings in the special Lagrangian and coassociative cases"
Hallo,
I am trying to read the paper "Calibrated embeddings in the special Lagrangian and coassociative cases" by Robert Bryant and I have a question. I hope that maybe one of you could give me some ...
- 339
1
vote
1
answer
361
views
Control of the $C^1$ norm of a diffeomorphism
Let $\Omega$ be a smooth open set of $\mathbb{R}^3$ diffeomorphic to the unit ball $B$. Let assumme that the boundary $\partial \Omega=\Sigma$ is also smooth and satisfies:
$$\int_\Sigma H^2 d\sigma \...
- 914
6
votes
0
answers
352
views
How to generate a random (Weyl) curvature operator ?
Given a dimension $n$, the space of curvature operators is the space $S^2_B(\Lambda^2\mathbb{R}^n)$ of symmetric endomorphisms $R$ of $\Lambda^2\mathbb{R}^n$ which satisfy the first Bianchi identity :
...
- 4,123
8
votes
1
answer
539
views
Known size invariant for Riemannian manifolds?
Larry Guth in his 2010 ICM address mentions the notion of a size invariant of Riemannian
metrics on a smooth manifold $M$. These are functions $S: Metrics(M) \to \mathbb{R}$ that are invariant under ...
- 81
1
vote
2
answers
341
views
Copies of ax+b inside the AN part of an Iwasawa decomposition?
As a relative novice to the structure theory of Lie algebras and Lie groups, the following is what I can gather from reading parts of Helgason's book DG, Lie groups and symmetric spaces and Knapp's ...
- 25.8k
7
votes
2
answers
518
views
Morse lemma with least amount of regularity.
I recently came across with $C^2$ Morse functions in my work and as I was reviewing some of the stuff I learned about Morse theory, I noticed that all the proofs of the Morse lemma I could come across ...
- 1,211
5
votes
1
answer
202
views
Forms satisfying the zero-energy condition on the projective plane
Theorem (Michel). A $1$-form on the projective plane is exact if and only if its integral over any projective line is equal to zero.
Is there a simple proof of this result due, I think, to R. Michel ?...
- 13.5k
7
votes
0
answers
1k
views
Closed geodesics on a closed, negatively curved Riemannian manifold
I have been searching for a while for a proof of the following fact: For a closed Riemannian manifold, all of whose sectional curvatures are negative, each free homotopy class of loops contains a ...
- 71
4
votes
1
answer
646
views
Combinatorial geodesics
[There has been a flaw in my definition - as Sergei and Andreas pointed out. I hope I could fix it.]
I want to understand how the concepts of directions, straight (or shortest) lines, and geodesics &...
- 9,720
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
6
votes
1
answer
468
views
Characterization of bounded geometry - Reference-request
I already asked this question at stackexchange three days ago. Since I got no answer, I want to try mathoverflow now. I hope that you can help.
I'm looking for a proof of an equivalence that can e.g. ...
- 63
20
votes
3
answers
2k
views
Non-stably trivial bundle with trivial characteristic classes
Though it's relatively clear that the characteristic classes do not characterise a vector bundle (and after looking through some books) I could not find an example of a vector bundle which is not ...
- 4,432
16
votes
2
answers
2k
views
There are two points on the Earth's surface that ... ?
At every moment in time, there are two points on the Earth's surface that have the same $\lbrace x, y, z, ... \rbrace$...?
What is the strongest, most impressive statement one can make here? The ...
- 151k
2
votes
2
answers
427
views
Analytic Lagrangian Submanifolds
Hallo,
I am looking for a preprint "Analytic Lagrangian Submanifolds" by Guillemin, Sternberg. I googled it but without any success. Does any one know how I could get this preprint. Or are there ...
- 339
0
votes
1
answer
339
views
Polarisation in a neighbourhood of a Lagrangian submanifold
Let $(X, \omega)$ be a symplectic manifold of dimension $2n$ and $\omega$ is an exact symplectic form i.e. $\omega = -d\alpha$. Let furthermore $M \subset X$ be a compact Lagrangian submanifold such ...
- 339
14
votes
1
answer
1k
views
Egg-ovoid rolling down an inclined plane
I am seeking a mathematical analysis of an egg-ovoid rolling down an inclined plane,
for pedagogical reasons.
It is well-known folk lore that the shape of an egg prevents it from rolling away from
...
- 151k
9
votes
2
answers
2k
views
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
15
votes
3
answers
4k
views
About MF Atiyah and R Bott's 1983 paper
I am a theoretical physics major student working on string theory. I want to understand the work of MF Atiyah and R Bott, "The Yang-Mills equations over riemann surfaces" . What kinds of mathematical ...
- 395
7
votes
1
answer
502
views
Fundamental groups of compact manifolds with non-negative Ricci curvature.
I would like to find an appropriate reference for the following statement:
Statement. Let $M$ be a compact Riemannian manifold with non-negative Ricci curvature.
Then $\pi_1(M)$ is virtually abelian.
...
- 14.3k
10
votes
1
answer
2k
views
How does the lack of partitions of unity affect the structure of analytic/holomorphic manifolds?
The standard way to define integration on a smooth manifold is to use partitions of unity, to extend to the case where the form you're integrating isn't supported on just one coordinate patch. Of ...
- 3,149
7
votes
1
answer
497
views
Open problems about CMC hypersurfaces with symmetries?
Recently, Andrews and Li announced a complete classification of CMC ($H=const.$) tori in $S^3$, confirming a conjecture of Pinkall and Sterling. Their main result is that any such torus is ...
- 5,891
24
votes
5
answers
4k
views
Weitzenböck Identities
I asked this question at Maths Stack Exchange, but I haven't received any replies yet (I'm not sure how long I should wait before it is acceptable to ask here, assuming there is such a period of time)....
- 19.3k
15
votes
2
answers
1k
views
When do blowups ''commute''?
Let $M$ be a manifold (variety, scheme, your favorite object) and let $N_1,N_2$ be two submanifolds (subvarieties, closed subschemes, ideal sheafes, etc.) such that $N_1 \cap N_2 \neq \emptyset$. ...
- 1,939
19
votes
1
answer
2k
views
Does this Banach manifold admit a Riemannian metric?
First, the question; after, the motivation.
Consider 27.6 (pdf pp. 262-263) in The convenient setting of global analysis (AMS, 1997), and, in particular, the example given at the end of it, which ...
- 7,831
17
votes
1
answer
526
views
Is $\partial X$ a sphere for $X$ a complete CAT$(0)$ space?
Let $X$ be a complete CAT$(0)$ metric space, and $\partial X$ its boundary.
One way to define $\partial X$ is as the equivalence class of geodesic rays
$\gamma(t), \gamma'(t)$
that remain within a ...
- 151k
13
votes
1
answer
2k
views
Convenient definition of "category of Riemannian manifolds"?
Has a notion of "category of Riemannian manifolds" been defined and used in the literature?
For which reasons is it or would it (not) be a useful notion?
I think the objects should be all (perhaps ...
- 23.3k
17
votes
4
answers
2k
views
Finite dimensional "Mountain Pass Lemma"
Question Does anyone know of a good reference which I can cite for the finite dimensional version of Mountain Pass Lemma?
Motivation I am writing a paper and found myself using the following result:
...
- 39.1k
7
votes
2
answers
1k
views
Kahler manifolds with constant bisectional curvature
It is well known that the universal covering of a complete Kahler manifold with constant bisectional curvature is $\mathbb{C}^n$, $\mathbb{B}^n$ or $\mathbb{CP}^n$. I need original paper(s) that prove ...
- 105
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
24
votes
1
answer
1k
views
Non-regular Connected Hausdorff Banach Manifold
After reading this MO post, I am wondering:
Is every (connected) Hausdorff Banach manifold a regular space?
Though unjustified, page 53 of this paper nonchalantly states: "Note that a Hausdorff ...
- 7,831
0
votes
1
answer
315
views
G-structures and complete riemannian manifolds
what are possible fundamental and introductory texts about G-structures ?
and where i can find the proof of this proposition:
if G(group) acts properly discontinuously on a space X , then G is a ...
- 165
4
votes
1
answer
479
views
Work on an Einstein-Hilbert type action but with the *absolute value* of scalar curvature?
This is only my second question on mathoverflow, so my apologies if this would be more appropriate at a physics site. My question concerns a modification to the Einstein-Hilbert action. The standard ...
- 175
10
votes
1
answer
2k
views
Curves of constant curvature on an ellipsoid
It is not difficult to see that the curves of constant geodesic curvature on a geometric sphere
are all circles: simple, closed curves that are geometric circles lying in a plane:
&...
- 151k
2
votes
1
answer
551
views
Heisenberg group: research themes
I am currently studying the Heisenberg group from the Riemannian geometry point of view, particularly focusing on its Gromov boundary and more generally its metric properties.
I would like to know ...
- 31