All Questions
Tagged with reference-request dg.differential-geometry
800 questions
3
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142
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Conjugacy classes of Cartan subspaces in parahermitian symmetric spaces
Are there any good tables of the numbers of conjugacy classes of Cartan subspaces in pseudo-Riemannian symmetric spaces? Or a good method to count them? In particular, I am interested in the ...
- 954
3
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0
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479
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de Rham currents/distributions on manifolds with boundaries
My main source for currents and distribution theory on manifolds in general is de Rham's Differentiable Manifolds. To recap, let $M$ be a smooth, $m$ dimensional real manifold without boundary. De ...
- 2,792
3
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247
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Fibre metrics on non-linear bundles
Usually what is meant under a fibre metric is that one is given a (smooth) vector bundle $\pi:Y\rightarrow X$, and on each fibre $Y_x$ an algebraic inner product $g_x$ that varies smoothly from point ...
- 2,792
3
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188
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References and results for the eigenvalues of Ricci tensor
I am looking for references or results that gives estimates for every eigenvalue of the Ricci tensor. For example, the least eigenvalue is related to the minimum of the Ricci curvature, what can we ...
- 1,774
3
votes
0
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159
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Upper bound on the geodesic distance in a Lipschitz domain
I was wondering if the following result is true. If yes, could you please suggest a reference. The result seems to have been used at several papers without quoting any reference. Is the proof ...
- 895
3
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0
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99
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Partial regularity of harmonic maps into spheres
Let $u : B_1(0) \subset \mathbb{R}^n \to S^k$ be an energy minimizing (minimizing in $H^1(B_1(0); S^k)$) harmonic map. I am trying to understand the theory of partial regularity, where the main claim ...
- 1,407
3
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336
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Understanding Calabi's conjecture proof: What is it meant by the logarithm of a differential form?
I'm reading several books and articles concerning Yau's proof of the Calabi conjecture. I want to have a deep understading of how and why such proof actually works, but most articles are aimed at ...
- 153
3
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0
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312
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Foliated vector bundle and basic connection
Let $(M,F)$ be a Riemannian foliation, i.e. there is a metric of $TM$ such that $g$ is bundle-like (locally $g_Q=g_{ij}(y)\,dy^i\otimes dy^j$ for a foliated chart $(x,y)$ and $Q=TM/F\cong F^\perp$).
...
- 1,915
3
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0
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114
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Jacobian of the action of a matrix on a Grassmannian
I'm looking for a reference concerning a calculation found in Furstenberg's 1963 paper "Non-commuting random products".
Lemma 8.8 of this paper states that if one takes a $d\times d$ invertible real ...
- 4,304
3
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0
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129
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Differential operators on a compact Lie group associated to bracket-generating sets
Let $G$ be a compact connected Lie group of dimension $m$ with Lie algebra $\mathfrak g$.
Let $\{X_1,\dots,X_h\}$ be a linearly independent set of $\mathfrak g$.
Assume that $\{X_1,\dots,X_h\}$ is ...
- 2,446
3
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135
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Asymptotic Expansion of Seiberg-Witten Differential?
Nekrasov & Okounkov proved (https://arxiv.org/pdf/hep-th/0306238.pdf) that the Seiberg-Witten prepotential can be given by
\begin{equation}
\mathcal{F}(\mathbf{a},\Lambda) = \lim_{\hbar\rightarrow ...
- 425
3
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348
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The uniqueness of Poincaré metric
The Poincaré metric $ds=\frac{\sqrt{dx^2+dy^2}}{y}$ has the proprety that the action of the group $PSL(2,\mathbb{R})=SL(2,\mathbb{R})/\{\pm I_{2}\}$ on $\mathbb{H}$ preserves the hyperbolic distance.
...
- 31
3
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112
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Is the square root of curl^2-1/2 a natural (Dirac-)operator?
In current computations on a particular $3$-dimensional Riemannian manifold, a first order differential operator $D:\Gamma^\infty(TM,M)\to \Gamma^\infty(TM,M)$ acting on vector fiels shows up, with ...
- 1,942
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74
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Deforming a non-positively curved Riemannian manifold into a negatively curved one
Cheeger deformations can be used to deform some non-negatively curved Riemannian manifolds into positively curved manifolds (e.g., sectional curvatures strictly positve), see
What is a Cheeger ...
- 1,942
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637
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English reference for Fischer-Grauert theorem and its generalization by Schuster
From this MSE question and its answer, and from this MO question I have learned of the following remarkable theorem of Wolfgang Fischer and Hans Grauert.
Theorem. A proper holomorphic submersion with ...
- 10.5k
3
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0
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141
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Is there a precisely formulable obstruction for the tangent bundle being a Lie algebra bundle?
Although vector fields (which are sections of the tangent bundle) form Lie algebras, the bundle itself, as far as I know, almost never carries Lie algebra structure; that is, in general, I believe, ...
- 18.4k
3
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203
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Reference: Differential geometry on surfaces that are graphs of 2D-fluid-equations and Point Vortices
In "Pressure Field, Vorticity Field, and Coherent Structures in
Two-Dimensional Incompressible Turbulent Flows" Larchev$\hat{e}$que computes the Gaussian curvature of the surface of the streamfunction ...
- 5,474
3
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0
answers
177
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Morse-Bott functions without critical manifolds of index 1 and n-1
I am now reading the article of M.F.Atiyah "Convexity and commuting hamiltonians" and I can't understand lemma 2.1. which says that if $\varphi \colon M \to \mathbb R$ is a Morse-Bott function without ...
- 2,305
3
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172
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question about currents
I have a question in the field of currents:
Let M be an n-dimensional smooth manifold, and let T be a k-current (induced by a k dimensional sub-manifold), I would like to approximate it by a series of ...
- 31
3
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0
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269
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covariant derivative of manifold-valued function and logarithm map
Let $M$ be a Riemannian manifold and $f\colon \Omega\subset \mathbb{R}^d\rightarrow M$ a smooth, i.e. $C^\infty$, function. For any $p\in M$ let $T_pM$ be the tangent space at $p$ and $\log_p\colon U\...
- 2,286
3
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71
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Transverse intersection in the $G$-orbit of paths
I know how to prove the following lemma but I assume that it is well-known. Can someone provide a reference for it?
Let $d>1$ and let $M$ be a $d$-dimensional connected smooth manifold with a ...
- 5,382
3
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0
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436
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does there exist a generalization of a manifold [closed]
Does there exist a generalization of a manifold whereby instead of being locally $\mathbb{R}^n$, it's locally another specified space?
3
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0
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153
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Gaussian heat kernel bounds on Riemannian manifolds [duplicate]
I wish to know if we have Gaussian lower and upper bounds for the heat kernel,i.e. $$
t^{-n/2}e^{-\frac{\rho(x,y)^2}{C_1t}} \lesssim p_t(x,y) \lesssim t^{-n/2}e^{-\frac{\rho(x,y)^2}{C_2t}},
$$
on a ...
- 31
3
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0
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406
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Smooth perturbation of a positive self-adjoint operator with compact resolvent
Consider a one-parameter family $A_t$ of unbounded positive self-adjoint operators with discrete spectrum (for example, one can consider a one-parameter family of Laplacians on a compact Riemannian ...
- 31
3
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0
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238
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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
3
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0
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256
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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
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447
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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
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159
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Strictly Convex Smoothing of a function defined on an affine manifold
A function defined on an interval is strongly convex if it has positive definite second derivative. A function on an affine manifold is strongly convex
if its restriction to each line is. An affine ...
- 700
2
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3
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613
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Manifolds with special holonomy especially $G_2$
I am interested in learning about $G_2$ manifolds and am aware that one of the canonical references is Joyce's Compact Manifolds with Special Holonomy. I am certain that my background is, at this ...
- 21
2
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2
answers
1k
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The existence of length-minimizing path between two points in a Riemannian manifold with boundary
Let $(M^n,g)$ be a Riemannian manifold with non-empty smooth boundary $\partial M$. For any two points $x,y\in M$, the distance between $x$ and $y$ may be defined as
$$ d(x,y)=\inf_\gamma Length(\...
2
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3
answers
478
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Groups of conformal isomorphisms of simply connected surfaces
By the uniformization theorem every connected and simply connected surface $M$ is conformally equivalent to one of the following three surfaces:
open disk $D$, complex plane $\mathbb{C}$, or $2$-...
2
votes
2
answers
389
views
Manifolds of continuous mappings.
I'm planning a short course on few topics and applications of nonlinear functional analysis, and I'd like a reference for a quick and possibly self-contained construction of a structure of a Banach ...
- 60.5k
2
votes
2
answers
2k
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Commuting of exterior derivative and contraction (vector-valued forms)
$\newcommand{\sig}{\sigma}$
$\newcommand{\tr}{\operatorname{tr}_{\eta}}$
$\newcommand{\al}{\alpha}$
$\newcommand{\be}{\beta}$
$\newcommand{\til}{\tilde}$
Let $E$ be a smooth vector bundle over a ...
- 6,741
2
votes
1
answer
963
views
Lie-derivative of tensor field along tensor field
What is the natural notion of the Lie-derivative of a tensor field along another tensor field, and where can I find an exposition of that?
- 5,327
2
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1
answer
320
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Flat scalar curvature on 4 manifold
Let $(M,g)$ be a closed(oriented) Riemannian 4 manifold. It is well-known that, if $scal^g\geq0$ and not identically zero, then $M$ admits a PSC metric by conformal transformation.
Q Is $T^4$ the ...
- 1,915
2
votes
1
answer
232
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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
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
2
votes
2
answers
411
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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
2
votes
1
answer
223
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Hodge decomposition for non-elliptic complexes
It is a well-known result that there is a bijective correspondence between harmonic sections and cohomology classes of an elliptic complex in a Riemannian/Hermitian manifold. Now consider Riemannian/...
- 167
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2
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523
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Orthogonal smooth vector field on a Riemannian manifold
Consider a compact Riemannian manifold $M$ with a smooth metric, and a smooth vector field $X$ on $M$. My question is, when can we construct another smooth vector field $Y$ on $M$ such that $Y$ is ...
- 1,407
2
votes
2
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357
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Set of vector fields with a particular expression of the commutator
I have a set $F$ of vector fields. The commutator $[v, u]$ is linear in $v$ and $u$ point by point, i.e., for each couple of vector fields $v\in F$ and $u\in F$, there are two scalars $a$ and $b$ such ...
- 469
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votes
1
answer
280
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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
1
answer
308
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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
2
votes
1
answer
1k
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Harmonic coordinates on Riemannian manifolds
I'm trying to read the paper of Jost and Karcher on the existence of harmonic coordinates on a ball whose size only depend on the injectivity radius and a two sided bound on the curvature.
...
- 4,123
2
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3
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397
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Reference request for structure equations
Let $(M,g)$ be a Riemannian manifold and let $\lbrace e_1,...,e_n\rbrace$ be a locally frame field on $M$ and $\omega _1 ,...,\omega _n$ be the dual $1$-forms of it. If $\omega _{ij}$ be the ...
- 601
2
votes
2
answers
305
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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}$$
Now pick up one such bundle $\xi$,we have the long exact sequence ...
- 111
2
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1
answer
338
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Reference on Deligne-Mumford compactness for Riemann surfaces
I am working with closed degenerating hyperbolic Riemann surfaces, and I try to understand the compactification of the moduli space. Looking in different books, notably the one of Hummel, I now get a ...
- 914
2
votes
2
answers
427
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Analytic Lagrangian Submanifolds
Hallo,
I am looking for a preprint "Analytic Lagrangian Submanifolds" by Guillemin, Sternberg. I googled it but without any success. Does any one know how I could get this preprint. Or are there ...
- 339
2
votes
1
answer
290
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On the stack of semistable curves
This is a question related to
Semistable curves of genus $g\geq 2$ form an Artin algebraic stack in the etale topology?
Let $\mathcal C\rightarrow \mathcal M^{ss}_g$ be the universal curve over the ...
- 494
2
votes
3
answers
336
views
For globally conformally flat surfaces, does radial symmetry of conformal factor imply the surface is a sphere?
We know that for the unit sphere in $\mathbb{R}^3$, the standard metric on such a sphere can be written as $$g=\frac{4}{(1+x_1^2+x_2^2)^2}(dx_1^2+dx_2^2)$$by a stereographic projection map from the ...
- 1,350