All Questions
Tagged with dg.differential-geometry riemannian-geometry
1,985 questions
5
votes
1
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411
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Averaging maps of Riemannian manifolds
Let $M$ be a compact Riemannian manifold. We know how to average functions $f\colon M\to {\mathbb R}$; the integral $\frac{\int_M f}{\int_M 1}$ returns a value in ${\mathbb R}$. If intead $f\colon M\...
- 666
11
votes
1
answer
600
views
Flat manifolds and irreducible representations
Let $M$ be a compact Riemannian manifold with vanishing curvature of Levi-Civita connection. Such manifolds were classified by Bieberbach; sometimes they are called Bieberbach manifolds. According to ...
- 9,237
12
votes
1
answer
1k
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How large can you draw an island on a map?
A cartographer friend asked me this question: could you classify (shapes of) islands by how much space they occupy on a map (comparatively to how much space is occupied by water) if you draw them as ...
- 1,347
4
votes
0
answers
97
views
Bi-Lipschitz classification of germs of conformal metrics at a singularity
First let me introduce some definitions.
By a germ of conformal metrics at a singularity, or simply a germ, I mean a conformal Riemannian metric $g$ defined on a punctured neighborhood $U$ of $0$ in $...
- 1,804
5
votes
1
answer
291
views
Gaussian Curvature of Exponentiated 2-Planes
Consider a Riemannian manifold $M$ with sectional curvatures $K\ge 0$ and let $\Pi$ be a 2-plane in the tangent space of $M$ at a point $p$. In a small enough neighborhood $U$ of 0 the exponential map ...
- 1,378
0
votes
1
answer
631
views
Green's function and eigenvalues with multiplicity
Green's function of a differential operator contains a lot of information of that operator. In particular, if we have a differential operator on a compact manifold with discrete spectrum, then Green's ...
- 587
6
votes
0
answers
830
views
Isometries of hyper-Kähler manifolds
For the purposes of this question, a hyper-Kähler manifold will be a complete connected Riemannian manifold $(\mathcal{M},g)$ whose holonomy representation is isomorphic to the natural representation ...
- 421
15
votes
1
answer
1k
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Thurston geometries in dimension 4
In the sense of W. Thurston here, there is 3 geometries in dimension 2 and there is 8 geometries in dimension 3.
Question: How many different geometries (in the sense of Thurston) do we have in ...
- 1,607
0
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2
answers
460
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Can simply or not simply connected maximally symmetric (Semi-)Riemannian manifold be completely classified?
A m-dimensional completed and connected (Semi-)Riemannian manifold which has $m(m+1)/2$ independent global Killing vector fields is called maximally symmetric space.
Then what are all possibilities ...
- 977
15
votes
1
answer
811
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Is the heat kernel more spread out with a smaller metric?
Suppose M is a smooth manifold, and we have two Riemannian metrics on M, say g and h, with g bigger than h (i.e. for every tangent vector at every point, the norm according to g is bigger than the ...
- 804
3
votes
0
answers
245
views
Question on a paper of Schoen and Yau
I am trying to understand the paper "Conformally flat manifolds, Kleinian groups and scalar curvature" by Schoen and Yau. In P.56, it says:
This implies that $\partial M$ has a zero $q$-capacity, ...
- 501
8
votes
2
answers
746
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Is there some Riemannian manifold's version of Whitney theorem?
Given any Riemannian or Semi-Riemannian manifold $(M,g)$, does there exist a Eucildean space $(E,g^\prime)$ of enough high dimension with metric $g^\prime=diag\{-1,-1,...,+1,+1,...\}$ with any n ...
- 977
3
votes
0
answers
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
3
votes
0
answers
280
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Is there any progress on Problem 13 (from Schoen and Yau)?
This is closed related to the question asked here. I wonder if there is any progress on Problem 13 from the "Problem Section" in Schoen and Yau, page 281, problem 13, which asks:
Let $M_1$ and $M_2$ ...
- 501
2
votes
0
answers
191
views
Generalized metric on spacetimes
I read many articles about space-times. Most authors consider these spaces as warped product manifolds $I\times M$ where $I$ is an open connected interval of the real line and $M$ is a Riemannian ...
- 422
23
votes
3
answers
5k
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Manifolds admitting flat connections
For each Riemannian manifold one can construct the Levi-Civita connection. While this connection is unique, we can call a (Riemannian) manifold flat if the Levi-Civita connection is flat. However when ...
- 9,330
4
votes
2
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306
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Reference for when a metric on a four-manifold is Kahler?
In a paper of Derdzinski1 (Proposition 5), he proved that if $\delta W^+=0$ and $W^+$ has at most two distinct eigenvalues, then the metric is (locally) conformally Kahler, and if in addition the ...
- 399
4
votes
1
answer
685
views
Horizontal lift of differential operator
On a Riemannian manifold $M$, there is a canonical horizontal lift $X^{\mathrm{hor}}$ of vector fields $X$ to $TM$, which is characterized by the two properties that
$X^{\mathrm{hor}}$ is a ...
- 9,853
0
votes
1
answer
408
views
Hilbert's Theorem relevance to positive curvature
In differential geometry, Hilbert's theorem (1901) states that there exists no complete regular surface S of constant negative Gaussian curvature K immersed in $ R^3 $. This theorem answers the ...
- 917
3
votes
1
answer
385
views
Does this squared distance functional have a unique critical point on geodesically convex manifolds?
Let $M$ be a Riemannian manifold with distance function $d$, $C \subset M$ a geodesically convex set, $a=(a_i)_{i=1}^n \in C^n$, $W \in \mathbb{R}_{\geq 0}^{n \times n}$ and $J\colon C^n \rightarrow \...
- 2,286
7
votes
1
answer
1k
views
Is the identification between symmetric tensors and homogeneous polynomials useful?
The general question:
Given an $n$-dimensional vector space $V$ over a field $k$, there exists an identification
$$\mathrm{Sym}^d(V) \sim k[x_1, \dots, x_n]_d$$
between the space of symmetric order $...
- 171
1
vote
0
answers
211
views
Riemann curvature of $S^1$-principal bundle
Let $(M,g)$ be a Riemannian manifold and $\pi:P \to M$ be $S^1$- principal fiber bundle endowed with a connection $\Gamma$. For every $p\in P$ we have,
$$T_pP \simeq T_pV\oplus\Gamma_p$$
Where $V$ ...
- 83
2
votes
0
answers
95
views
Normal-like coordinates for weakly differentiable metrics
Let $(M,g)$ be a Riemannian $W^{2,p}$ metric, with $p>n/2$. Thus $g$ is at least continuous. At any point $P\in M$, do there exist local coordinates $x^i$ such that $g$ can be decomposed as $g_{ij} ...
- 83
2
votes
2
answers
613
views
Approximation theorem for Anti-Self-Dual Metrics
Rounge's Theorem states that any meromorphic function on a domain inside $\mathbb{C}$ can be approximated (over compact subsets) by a sequence of rational functions (meromorphic functions on $\mathbb{...
- 17.5k
8
votes
2
answers
773
views
The necessary and sufficient condition for $\textbf{global}$ conformal flatness of a n-dim (pseudo-)Riemannian manifold
There is a theorem :
1) 2-dim (pseudo-)Riemannian manifold must be local conformal flat;
2) 3-dim (pseudo-)Riemannian manifold is local conformal flat iff the Cotton tensor vanishes.
3) n-dim (n>3) ...
- 977
4
votes
0
answers
299
views
Focal points for the exponential map and Jacobi fields
It is known that in a Riemannian manifold $(M,g)$, if there is a closed geodesic and a non-zero, periodic, non-constant Jacobi field along it, then M has a focal point. Is the converse true? That is ...
- 41
3
votes
0
answers
556
views
connections and curvature
Let $(M, g)$ be a Riemannian manifold. Is it possible to construct two different affine (or metric) connections, say $\nabla$ and $\nabla'$, which induce the SAME curvature tensor, i.e. $R(X, Y)Z=R'(...
- 1,219
6
votes
1
answer
1k
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Laplace-Beltrami operator on a Lie group
For an arbitrary Lie group, is it always possible to chose a left-invariant Riemannian metric such that the Laplace-Beltrami operator $\Delta$ is given by
$$\Delta f = \delta^{i j} X_i X_j f$$
for ...
- 6,025
3
votes
1
answer
326
views
Spectral multipliers vis-a-vis Differential geometry
Let us mention two papers for examples: this one by Seeger and Sogge and this by Cheeger, Gromov and Taylor. One can also mention papers by Stein, for example, this one. There are also many others of ...
- 141
0
votes
1
answer
590
views
Yang-Mills equations are not elliptic [closed]
How does one prove that the Yang-Mills equations (from classical Yang-Mills theory) are not elliptic?
Alternatively, how does one calculate the principal symbol of the Yang-Mills equations?
Can ...
- 376
18
votes
2
answers
2k
views
If there is a dense geodesic, are almost all geodesics equidistributed? Dense?
Let $M$ be a complete finite volume Riemannian manifold and $\gamma : \mathbb{R}^{\geq 0} \to M$ a geodesic. Suppose that $\mathrm{im}(\gamma)$ is dense. Is it equidistributed in the Riemannian ...
- 13.8k
7
votes
1
answer
1k
views
The surjectivity of the exponential map for the isometry group
Little is known on general conditions guaranteeing that the exponential map between a Lie algebra and an associated Lie group is surjective.
Let $M$ be a noncompact connected Riemann manifold, and $G$...
- 5,407
4
votes
1
answer
566
views
The Chern connection on a Hermitian symmetric domain
There's a connection (the Chern connection) on the Tangent Bundle of a Kahler Manifold which is compatible with both the hermitan metric, and the holomorphic structure. In general, I guess there's no ...
- 191
26
votes
2
answers
2k
views
Ellipses on spheres (and other surfaces)
Define an ellipse $E$ on a sphere as the locus of points whose sum of
shortest geodesic distances to two foci $p_1$ and $p_2$ is a constant $d$.
There are conditions on $\{ p_1, p_2, d \}$ for this ...
- 151k
2
votes
1
answer
367
views
The limit of a sequence of embedded minimal disks in $\mathbb{R}^3$
Let $\Sigma_n,n\ge 1$ be a sequence of embedded minimal disks in $\mathbb{R}^3$ such that:
(1) $0\in\Sigma_n\subset B(0,r_n)$ with $r_n\to\infty$ as $n$ tend to $\infty$,
(2) $\partial\Sigma_n\...
- 23
2
votes
1
answer
246
views
The points of half area of a triangle
Let $S$ be a simply connected Riemannan surface . Suppose $\Delta ABC$ is a triangle on $S$. The Area of a triangle is denoted by $\mathcal{A}$. A point $P$ in the interior of $\Delta ABC$ is ...
- 356
5
votes
1
answer
327
views
What is the difference between $\delta W^{\pm}=0$ and Einstein?
Maybe this is a vague question. In Besse's book Einstein manifolds, $\delta W^{\pm}=0$ is considered as a generalization of Einstein metrics on four-manifolds. I was wondering what is the difference ...
11
votes
1
answer
584
views
Decomposition of $\mathrm{O}(n)$-modules coming from differential geometry
Let $V$ be a $n$-dimensional real vector space equipped with a positively definite scalar product $g$ and let $\mathrm{O}(n)$ be the automorphism group of $(V,g)$. View $V^{\otimes k}$ as a $\mathrm{O}...
- 1,804
6
votes
1
answer
392
views
Analytic representatives for Kahler classes
If we are given compact complex manifold $X$ and a Kahler class $[\omega]$,
can we always find a positive definite representative $\omega \in [\omega]$ that is
real analytic?
- 1,727
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
3
answers
1k
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Is the group of isometries of a homogeneous Riemannian manifold maximal?
I have a homogeneous Riemannian manifold X with isometry group Iso. Is Iso a maximal group? By maximal group, I mean that there does not exist another group G such that:
Iso is a proper subgroup of G,...
- 71
2
votes
0
answers
149
views
Variational inequality on Manifold
Let $(M,g)$ be a Riemannian manifold. Consider $A : W^{1,r}(M,\mathbb{R}) \rightarrow W^{-1,r'}(M,\mathbb{R}), k \mapsto Ak$, where $Ak$ is defined by $(Ak)(\varphi) = \int_{M}g(\nabla k, \nabla \...
- 21
3
votes
1
answer
340
views
A question on Schrodinger operator
I am not sure whether I should ask for help here or math stackexchange. I got trouble with an inequality involving the Schrodinger operator on manifolds. Any suggestion is appreciated!
Let $(M,g)$ be ...
2
votes
1
answer
338
views
Volume bounds of balls in Riemannian manifolds
Let $(M,g)$ be a complete Riemannian manifold and suppose $\mathrm{Ric}(g) \geq -k$ for some $k>0$. Suppose we know that $\mathrm{vol}_g (B_1^g (x_0)) \geq \nu$ for some particular $x_0 \in M$ and ...
- 21
7
votes
1
answer
2k
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Volume of geodesic balls
I have two questions (somewhat related) regarding local geometry on a SMOOTH, COMPACT Riemannian manifold. I still have a hard time getting a "good" understanding of local geometry.
Question 1:
It ...
- 207
4
votes
1
answer
587
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Counterexample to volume comparison inequality assuming only scalar curvature bound?
The Gromov-Bishop volume comparison theorem says that if we have a lower bound for the Ricci curvature on $(M,g)$, then its geodesic ball has volume not greater than the geodesic ball with the same ...
- 141
3
votes
0
answers
198
views
$\mathbb{CP}^1$-structures and hyperbolic Gauss maps
Let $\Sigma$ be a closed surface of genus at least $2$.
Put a quasi-Fuchsian $\mathbb{CP}^1$-structure (i.e. complex projective structure) on $\Sigma$. Thus the universal cover $\tilde{\Sigma}$ is ...
- 1,804
6
votes
1
answer
2k
views
Taylor expansion of the determinant of a Riemannian metric
Let $(M,g)$ be a compact Riemannian manifold without boundary. Fix a point $x\in M$ and $N\ge 2$ large. Then there exists a metric $\tilde g$, conformal to $g$ such that $$ \det \tilde g=1+O(r^N)$$ ...
- 327
3
votes
1
answer
928
views
Estimate the smallest eigenvalue of a Schrodinger operator
There are several results on the estimate of the number of negative eigenvalues of a Schrodinger operator, see a recent paper of Grigor'yan-Nadirashvili-Sire and references therein. I wonder how to ...
- 399
0
votes
1
answer
287
views
Describe all differentiable functions on $\mathbb{S}^n \backslash S$ (S is the south pole) [closed]
Consider the sphere $\mathbb{S}^n$ embedded in $\mathbb{R}^{n+1}$. Let $N$ be the north pole of the sphere and $S$ the south pole. Every point on $\mathbb{S}^n \backslash \{N,S\}$ is defined uniquely ...
- 303