All Questions
Tagged with reference-request dg.differential-geometry
800 questions
8
votes
1
answer
1k
views
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of the eigenforms and spectrum of the Laplacian acting on $p$-forms on the round sphere $S^n$, apparently citing an unpublished computation of ...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
4
votes
1
answer
532
views
Diffusion semigroup generated by Laplacian
Let $M$ be a complete Riemannian manifold and $\Delta$ denote the Laplacian on it. Also assume that the spectrum of $-\Delta$ lies inside $[a, \infty)$. Let $P_t, t > 0$ denote the diffusion ...
- 43
6
votes
2
answers
2k
views
How many metrics of constant curvature exist on a Riemannian surface?
I have been trying to determine the number of metrics of constant curvature on a surface of genus $n$, say $\Sigma$. For low values, the answer is clear, the moduli space is a point for the sphere, ...
- 841
2
votes
1
answer
232
views
Shortest paths in Alexandrov spaces
Let $X$ be an Alexandrov space with curvature bounded from below (if necessary, $X$ might be assumed to be finite dimensional or even compact).
Question 1. Is it true that every point of $X$ has a ...
- 21.8k
1
vote
3
answers
502
views
orbit space of $\mathbb{Z}_p$ action over complex projective space by permuting the homogeneous coordinates
$Z_p$:=cyclic group of order $p$.
I want to understnd $H_\ast(\mathbb{C} P^{n}/Z_{n+1};Z)$ with $(n+1)$ being a prime number,and the action is given by permuting the homogeneous coordinates.
For ...
- 593
7
votes
1
answer
1k
views
Differential forms along the fiber
Let $E \to M$ be a smooth fiber bundle. Instead of differential forms defined on the whole tangent bundle $TE$ one could also consider forms on the vertical tangent bundle $VE$, i.e. forms defined on ...
- 5,824
1
vote
1
answer
1k
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Hermitian form, fundamental $2$-form of Kahler structure on $\mathbb{C}^n$
I've come across the following (it is an excerpt of Stolzenberg's lecture notes 19):
Wirtinger's Inequality.
Let $L$ be a complex linear space and let $M$ be a real
even-dimensional subspace....
- 103
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
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
1
vote
1
answer
265
views
Boundary components of a subsurface
Consider the following situation. Suppose we have a closed oriented Riemannian surface $ \Sigma $ and a connected open subset $ \Omega \subseteq \Sigma $ with a boundary, consisting of finitely many ...
- 337
57
votes
7
answers
8k
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Maryam Mirzakhani's works
Maryam Mirzakhani has made several contributions to the theory of moduli spaces of Riemann surfaces.
Mirzakhani was awarded the Fields Medal in 2014 for "her outstanding contributions to the dynamics ...
user68866
1
vote
0
answers
224
views
Characterization of the Riemann curvature tensor [duplicate]
Let $(M^n,g)$ be a Riemannian manifold, $a\in M$ be a fixed point. It it well known that there exists a coordinate system near $a$ (e.g. the normal one) such that
$$g_{ij}(x)=\delta_{ij}+O(|x|^2).$$
...
- 21.8k
15
votes
2
answers
1k
views
Is there a citeable reference for star-shaped open subsets of R^n being diffeomorphic to R^n?
A folk theorem says that star-shaped open subsets of R^n are diffeomorphic to R^n.
Is there a citeable reference for a proof of this result?
For the sake of being definite, let's say that
“citeable” ...
- 37.8k
2
votes
1
answer
280
views
Unitary representation with fixed Casimir
Let $G$ be a connected reductive real Lie group with Lie algebra $\mathfrak{g}$. We denote by $\widehat{G}_u$ the unitary dual, that is the set of isomorphism classes of unitary reprensentation of $G$....
- 1,111
2
votes
2
answers
305
views
boundary homomorphism in the homotopy exact seqeunce of principal $SO(9)$ bundle over $S^8$
Consider principal $SO(9)$ bundles over $S^8$.They are in 1-1 correspondence with $$[S^8,BSO(9)]\cong \pi_7(SO(9))\cong \mathbb{Z}$$
Now pick up one such bundle $\xi$,we have the long exact sequence ...
- 111
6
votes
2
answers
2k
views
Line bundles over Kähler–Hodge manifolds
A Kähler–Hodge manifold $M$ can be defined as a Kähler manifold whose Kähler form $\omega$ is integral, namely $\omega\in H^{2}(M,\mathbb{Z})$. It is known then that there always exists a Hermitian ...
- 2,816
23
votes
2
answers
1k
views
fake $S^{2k}\times S^{2k}$
Let $X$ be a fixed closed manifold,$S(X)$ the structure set and $Aut(X)$ the group of self homotopy equivalence of $X$.
surgery theory tells us that $\mathcal{M}(X):=S(X)/Aut(X)$ is in bijection ...
- 593
12
votes
1
answer
727
views
Schemes over topological rings
I have recently been interested in studying an extension of 'usual' algebraic geometry to take into account the topology of $R$ in the definition of the affine scheme $\mathrm{Spec}\, (R)$ when the ...
- 1,270
0
votes
1
answer
201
views
Ambient isotopy of the diagonal submanifold in product space
Given a closed manifold $M^n$ and its $k$-fold product space $M^n\times\cdots\times M^n$,Can the diagonal submanifold $\Delta:=\{(m,\cdots,m)\in (M^n)^k\mid m\in M\}$ be isotopied to the submanifold
$...
- 73
6
votes
2
answers
1k
views
Shuffle (co-)multiplication and generalized Leibniz formula in tensor calculus
The headline already says it: Is anybody (except me, UPDATE: plus Gavrilov) aware of this formula for higher total covariant derivatives of tensor products?
It is the simplest application of the ...
- 1,068
5
votes
1
answer
482
views
Besse p134 Riemann tensor in dimension 4
Does someone have a reference for the proof of 4.72 page 134 of Einstein Manifolds? It is said that
$$\check{R}-\vert R\vert^2g/4=S/3 (Ric-S/4) +2\mathring{W}(Ric -S/4) $$
because we are in dimension ...
- 914
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
12
votes
1
answer
1k
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What is a good introduction to cluster algebras from surfaces?
What is a good reference for cluster algebras from surfaces, with a view to their connection to Teichmuller theory?
In my view, that means it should start off with unpunctured surfaces (and in fact,...
- 6,292
6
votes
2
answers
446
views
Is every $S^3$ block bundle over $S^4$ a fiber bundle?
I am interested in the difference between block bundle and fiber bundle.
Let $K$ be a simplicial complex and $p: E\to |K|$ be a continuous map.
A block diffeomorphism of $\Delta^p\times M$ is a ...
- 73
1
vote
1
answer
409
views
Heat kernel upper bound on compact Riemannian manifold
Let $M$ be a compact Riemannian manifold without a boundary. Let $p_t(x,y)$ be the heat kernel. I am looking for a reference for the result: there exists a constant $C$ such that
$$|p_t(x,y)| \leq C$$
...
- 65
5
votes
0
answers
264
views
Continuity of the curve-shortening flow with respect to the curve
The curve-shortening flow is an evolution equation for a smooth closed curve $\alpha$ inmersed in a Riemannian surface $M$. The version where $M$ is the Euclidean plane is illustrated for example in ...
- 4,304
1
vote
2
answers
139
views
Reference request: minimal (maximal) Lorentzian surfaces in $\mathbb{R}^{1,2}$
Let $R^{1,2}$ be the Minkowski 3-space, I would like to know any references about minimal (maximal) orientable Lorentzian surfaces in $\mathbb{R}^{1,2}$, including examples and maybe general theories, ...
- 783
3
votes
0
answers
238
views
Parallel Ricci condition - Status report and bibliography
First I'd like to point out that I'm not a mathematician but a physicist. Dealing with a (hopefully) new affine theory of gravity we have find that the equation of motion are not the usual Einstein's ...
- 690
0
votes
1
answer
182
views
Surjectivity of "nice maps" from local properties
What tools are available from real algebraic geometry, analysis and
topology to check surjectivity of a map $f:M_{1}\rightarrow\mathbb{R}^{d}$
from local properties and maybe function values?
...
- 1,256
2
votes
2
answers
415
views
Geometrical interpretation of a Schrödinger operator
Consider a $2 \times 2$ Hermitian (or symmetric) matrix-valued function
$$g(x) = \{ g_{jk}(x)\}_{j,k=1,2}, \quad x \in \mathbb{R}^{2},$$
such that $0 < m_{-}I \leq g(x) \leq m_{+}I$, for some $m_{-}...
- 321
1
vote
2
answers
314
views
Notion of manifold curvature?
Consider a particular embedding of a $C^2$ manifold $\mathcal{M}\subseteq\mathbb{R}^m$. Given $p\in\mathcal{M}$, suppose $\epsilon>0$ is small enough that the portion $U$ of $\mathcal{M}$ which is ...
- 7,633
1
vote
1
answer
165
views
Can any bounded area defined by polynomial inequality in $\mathbb{R}^n$ be partitioned into simply connected finite area such that
Can any bounded area defined by polynomial inequality in $\mathbb{R}^n$ be partitioned into simply connected finite areas such that for each simply finite area there exist a diffeomorphic map that ...
- 3,094
6
votes
2
answers
701
views
Lower regularity version of Moser's theorem on volume elements
A theorem of Moser, published in "On the Volume Elements of a Manifold" (Transactions of the Americal Mathematical Society 120, 1965; doi: 10.1090/S0002-9947-1965-0182927-5, jstor), shows that if a $C^...
- 23.2k
4
votes
0
answers
241
views
"Partition" of a smooth function in $\mathbb R^2$
This is a question asking for reference.
I have a proof of the following.
Let $f=f(x,y)$ be a smooth function in $\mathbb R^2$ which vanishes at the origin. Then there exist smooth functions $f_1=...
- 231
3
votes
0
answers
256
views
Uniqueness of scalar curvature
I'm reading Gromov's notes
http://www.ihes.fr/~gromov/topics/SpacesandQuestions.pdf
and at page 7 they say that there is a unique second order differential operator $S$ from the space of Riemannian ...
- 585
1
vote
2
answers
188
views
Calculating Exterior Distance from Measurements of Inner Geometry
Gauss has proven in his famous Theorema Egregium, that it is possible, to calculate the gaussian curvature from measuring angles and distances on the surface, irrespective of how the surface is ...
- 13.2k
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
4
votes
0
answers
273
views
An example of mean curvature flow that does not preserve embeddedness
Let $F: M^n \to \mathbb R^{n+k}$ be an embedding and $F_t$ be a families of immersions so that $F_0=F$ and
$$\frac{\partial F_t}{\partial t} = \vec H$$
It is known that in hypersurface case ($k=1$),...
- 534
10
votes
2
answers
876
views
Are there nontrivial involutions of $S^7\times S^7$ with fixed point set homeo to $S^7$?
The group $\mathbb{Z}_2$ acts on $S^7\times S^7$ by switching the coordinates with fixed point set $\Delta(S^7\times S^7)\cong S^7$. I want to know whether there are some other $\mathbb{Z}_2$ actions ...
- 888
2
votes
0
answers
179
views
Questions about transformation or integral transformation
I have asked several mathematicians about the following questions,but all of them think they are good questions,but can not give a complete answer.Now I have to come here to ask mathematicians all ...
- 3,094
5
votes
0
answers
310
views
Reference for Hodge decomposition
Let $U$ be a bounded open subset of $\mathbb{R}^d$ with Lipschitz boundary, and $g \in L^2(U,\mathbb{R}^d)$ be a solenoidal vector field (i.e. $\nabla \cdot g = 0$). Then $g$ can be written in the ...
- 562
13
votes
0
answers
872
views
Geometric meaning of the black hole horizon
It is widely accepted that the singularity of the Schwarzschild metric at the event horizon is purely an artifact of the coordinates and no physical singularity exists at the horizon. However, as ...
- 16.5k
7
votes
3
answers
968
views
Affine structures
I would like to study manifolds endowed with a linear connexion $\nabla$ which is torsion free and locally flat i.e. its curvature is $0$ (such a connexion is called flat if in addition, its holonomy ...
- 442
4
votes
1
answer
711
views
Reference request for instantons
I've been researching instantons lately and I'd like to learn more about them but would like some help finding what to read. I have read about the ADHM equations and their noncommutative analogues. ...
- 161
3
votes
0
answers
447
views
Complex structures on Riemann surfaces
This is cross posted from math.SE: https://math.stackexchange.com/q/876432/9
Let $M$ be a Riemann surface and $[\alpha] \in H^{0,1}(M; T^{1,0} M) \simeq (H^0(M;K^2))^*$. Considering $\alpha$ as a ...
- 3,244
10
votes
3
answers
2k
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nth term in the Baker-Campbell-Hausdorff formula
I am trying to prove a result for which I need the nth term of the Baker-Campbell-Hausdorff formula. I came at this particular result (which is not of significance for the question, but mentioning for ...
- 245
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
7
votes
1
answer
428
views
A geometric characterization of smooth points of a complex algebraic variety
Let $X^m\subset \mathbb{C}^n$ be an irreducible $m$-dimensional complex algebraic subvariety. Let $\mathbb{C}^n$ be equipped with the standard Hermitian metric.
Fix an arbitrary point $p\in X$. Let $...
- 21.8k
7
votes
2
answers
2k
views
The integral of torsion
I found the following * exercise (exercise *9) in page 407 of the book of do Carmo "Differential geometry of curves and surfaces". This problem is a classical theorem which is referenced ...
- 356
7
votes
0
answers
116
views
Bundles over Function Spaces
Is there any reference on bundles over function spaces? In particular, I am interested in Banach-bundles over function spaces like $W^{k,r}(M)$, where $M$ is a Riemannian manifold. Separable Hilbert-...
- 71