Questions tagged [riemannian-geometry]
Riemannian Geometry is a subfield of Differential Geometry, which specifically studies "Riemannian Manifolds", manifolds with "Riemannian Metrics", which means that they are equipped with continuous inner products.
2,971
questions
5
votes
1
answer
1k
views
Eigenforms for Laplacian on a non-flat two-torus
Does anyone know an explicit, exact description of the eigenforms of the Laplacian on a non-flat two-torus?
- 3,257
9
votes
1
answer
568
views
Injectivity radius of the Sasaki metric
Let $(Q,g)$ be a (compact) Riemannian manifold with injectivity radius $\rho>0$. There is a natural metric $\tilde g$ on the tangent bundle $TQ$ which is known as the Sasaki metric and which makes $...
- 125
5
votes
0
answers
1k
views
"The famous Lusternik-Schnirelmann Theorem of the Three Closed Geodesics"
The title is a quote from p.256 of Wilhelm Klingenberg's 1995
Riemannian Geometry (Google Books link):
Every surface homeomorphic to a sphere $\mathbb{S}^2$ has three distinct, simple, closed ...
- 149k
7
votes
2
answers
1k
views
Pólya's conjecture on the spectra of the Laplacians
Recently I've learned something about the spectra of the Laplacians. Given a bounded domain $\Omega \subset \mathbb{R}^n$ with $\partial \Omega$ smooth, we can consider eigenfunctions of Dirichlet ...
- 355
2
votes
1
answer
303
views
2 questions about loops and negative curvature
$(M^n,g)$ is a compact $n$ dimensional manifold of negative curvature with n>2 . let $\alpha$ be a simple closed geodesic loop in $M$ based at a point $p$
1) will the geodesic in the free homotopy ...
student
5
votes
1
answer
333
views
Local splitting of the tangent bundle with interesting properties
Let $(M,g)$ be a Riemannian manifold and let $U\subset M$ be an open subset. Suppose that the tangent bundle over $U$ splits into two orthogonal distributions $TU=\mathcal{E}\oplus \mathcal{F}$.
Is ...
- 688
5
votes
0
answers
267
views
stochastic control / geometric mean
Consider the following problem:
Given $\Omega$ and $U$ two symmetric definite positive matrices, choose a matrix $K$ to minimize the expectation $x' \Omega x + x'K'UKx$ when $x$ follows the invariant ...
- 427
1
vote
2
answers
1k
views
Geometric Mean Value Property
Does anyone know where I could find a proof of a variant of a version of the mean-value property for harmonic functions in Riemannian manifolds? I'm actually more interested in using an elliptic ...
- 11
1
vote
1
answer
393
views
Holonomy group of cotangent bundle
Is the holonomy group of the cotangent bundle, of a compact riemannian manifold, with respect to te standard symplectic structure equal to $SU(n)$, where $n$ is the dimension of the riemannian ...
- 13
8
votes
2
answers
959
views
Chebyshev net in 3D
I would like to know the reasons why the existance of Chebyshev net in 3D-case is problematic.
This question boils down to the PDE described below.
(I do not know much about PDEs, so feel free to say ...
- 43.7k
18
votes
1
answer
944
views
Possible isometries of a positively curved $S^2\times S^2$
Just to put things in perspective, recall that the Hopf Conjecture asks whether $S^2\times S^2$ admits a metric of positive sectional curvature. By the work of Hsiang-Kleiner, it is known that, if $S^...
- 5,831
6
votes
2
answers
2k
views
Metric associated to a Connection on a Vector Bundle
General question: Given a vector bundle $E \rightarrow M$ on a complex manifold $M$, and a connection $\nabla$ on $E$, is it possible to find an Hermitian structure on $E$ such that $\nabla$ is the ...
9
votes
1
answer
562
views
Length spectrum for Riemannian metrics in the projective plane
Are there (known) examples of non-isometric Riemannian metrics on the projective plane that have the same length spectrum?
This question is related to MO questions Length spectrum and Zoll surfaces ...
- 13.2k
3
votes
1
answer
400
views
Length spectrum and Zoll surfaces of revolution
The earlier MO question, "Length spectrum of spheres," asked if the length spectrum of closed
geodesics determines the metric on $S^2$, and the answer was a clear No due to Zoll surfaces,
all of whose ...
- 149k
10
votes
0
answers
378
views
Is it overkill to invoke Kirszbraun theorem to prove the following fact ?
Given a small enough convex triangle $(abc)$ in a (smooth or Alexandrov) surface $(X,d)$ of curvature greater than $-1$, let $(\overline{abc})$ be its comparison triangle in $\mathbb{H}^2$. Then there ...
- 4,033
12
votes
1
answer
2k
views
Integration By Parts on Non-compact Manifolds
This is undoubtedly a very easy question, but perhaps there are some subtleties. Under what circumstances can we integrate by parts over a non-compact Riemannian manifold? I am aware that having ...
- 1,113
7
votes
2
answers
923
views
Full isometry groups of Stiefel and Grassmann manifolds
Hi,
I'm looking for a reference for the full isometry groups of the
(i) complex Stiefel manifolds $U(m)/U(m-l)$, either for the Euclidean metric (i.e. identifying it with orthonormal $m \times l$-...
- 2,926
7
votes
3
answers
909
views
Conformal structure does not see conical singularities
the conformal structure does not see the conical singularities of a polyhedral surface.
This is a quote from the Preface of Quantum Triangulations (eds.: Carfora, Marzuoli).
The sentiment is ...
- 149k
14
votes
0
answers
695
views
Best metrics on exotic R^4
What is known about the existence of complete metrics with good properties (e.g., Einstein, constant scalar curvature, etc...) on exotic ${\bf R}^4$s? Note, that some exotic ${\bf R}^4$s have non-...
- 385
2
votes
1
answer
910
views
What's the relationship between the Riemannian metric and Jacobi field?
I encounter to the question in reading the following Excise:
Let $(M,g)$ be a $m$-dimensional Riemannian manifold, and $(r,\theta^1,\theta^2,\dotsc,\theta^{m-1})$ be the (geodesic) polar coordinate. ...
- 155
17
votes
5
answers
2k
views
Is there a coordinate-free proof of the hamiltonian character of the geodesic flow?
I do not know if this question is appropriate for this site, but I posted here without having answers, so now I make this attempt.
Let be $(M,g)$ a pseudoriemannian manifold. Let us identify the ...
- 4,266
6
votes
1
answer
3k
views
Fubini Study Metric and Einstein constant
Hi all,
it is well known that the complex projective space with the fubini study metric is Einstein, but what is the explicit value, i.e. for which $\mu$ does $Ric=\mu g$ hold?
Moreover, I would ...
- 688
1
vote
0
answers
300
views
einstein metrics on the tangent bundle
hi,
i have the following question. let $M$ be a compact, real analytic, riemannian manifold with real analytic metric $g$. does the tangent bundle admit always a einstein metric ?
marco
- 213
4
votes
0
answers
616
views
Cheeger's Finiteness Theorem and Lipschitz Constant
Cheeger's Finiteness Theorem states that
For each positive numbers $D,v,n$, the
number of diffeomorphism classes of Riemannian manifolds $M$ with
$Diameter(M)\le D$, $Vol(M)\ge v$, and $|K(M)|\le 1$ ...
- 1,101
3
votes
0
answers
269
views
Seek "typical examples" for the structure of spaces with two-sided Ricci bounds
By a 1990 paper of Michael Anderson, the following is true:
Theorem. Let the metric space $(X,d,p)$ be a pointed Gromov-Hausdorff limit of a sequence of complete pointed Riemannian manifolds $(M_i,...
- 3,192
4
votes
4
answers
3k
views
space of geodesics
hallo,
i have the following problem: Let $(M,g)$ be a compact Riemannian manifold with metric $g$ and $\nabla$ be the Levi-Civita Connection. Denote by $G(M) =${$\gamma: \mathbb{R} \rightarrow M | \...
- 213
6
votes
1
answer
1k
views
How the Jacobi metrics may be useful in mechanics with or without constraints?
A mechanical system $(Q,K,V)$ is specified by the configuration space $Q,$ the potential energy $V\in C^\infty(Q),$ and the kinetic energy $K=K_g$ given by a Riemannian metric $g$ on $Q.$
If $V{<}...
- 4,266
5
votes
1
answer
802
views
Partitions of Unity
Fix a metric $g$ on a smooth, closed manifold $\mathcal{M}$. Take a finite subcover of the manifold from its atlas. Is it true that any smooth partition of unity subordinate to this cover has ...
- 1,113
10
votes
3
answers
2k
views
Examples and non-examples of Riemannian foliations
Recall a tranverse metric on a (regular) foliated manifold $(M,F)$ is a positive symmetric $C^\infty (M)$-bilinear form $g$ such that
1) $Ker(g_x)=T_x F$
2) It is invariant with respect to lie ...
5
votes
2
answers
929
views
Which vector bundle are the Christoffel symbols sections of?
The collection of Christoffel symbols $\Gamma_{ij}^k$ of a connection (or of a metric) on a smooth manifold $M$ is not the collection of components of a tensor field in some local chart, i.e. they don'...
- 22.7k
5
votes
1
answer
356
views
Symmetries vs. Bound in codimension of Nash isometric embedding
Let $(M^m,g)$ be a compact smooth Riemannian manifold of dimension $m$. From the celebrated Nash Embedding Theorem, we know there exists a (smooth) isometric embedding $M\hookrightarrow\mathbb R^n$ on ...
- 5,831
6
votes
2
answers
3k
views
Metric Connections on a Lie Group
A Lie group has three standard Cartan connections; the (-)-connection, the (0)-connection, and the (+)-connection. The (0)-connection is Levi-Civita with the associated metric the bi-invariant metric. ...
- 1,368
12
votes
2
answers
2k
views
Existence, uniqueness, and regularity for linear parabolic PDE on a complete Riemannian manifold
Let $M$ be a smooth manifold with a complete Riemannian metric $g$ and $E$ a smooth vector bundle over $M$ with an inner product and compatible connection $\nabla$. Let $K: E \rightarrow E$ be a ...
- 27k
6
votes
0
answers
360
views
Compactness of solutions to parabolic equations (parabolic regularity)
I am working on a Ricci flow $(\mathcal{M} , g(t))$, for which the conjugate heat operator is $\Box ^* := - \partial _t - \Delta + R$, where $R$ is the scalar curvature.
For each $s>0$, I have a ...
- 1,113
3
votes
1
answer
434
views
eigenspinors of Dirac operator
$M$ compact manifold. Let $\lambda$ be an eigenvalue for the Dirac operator of multiplicity greater than 2. I'm interested in showing the existence of two linearly independant eigenspinors $u$ and $v$ ...
- 119
12
votes
4
answers
2k
views
Riemannian metric on a flag variety
$\def\C{\mathbb{C}}\def\CP{\mathbb{CP}}$Every complex projective space $\CP^n$ has a natural Riemannian metric, the Fubini–Study metric, which is defined via the quotient definition of $\CP^n = \...
- 7,183
4
votes
0
answers
241
views
Methods for generating metrics and minimizing variational dynamics of particles (masses or charges) on n-dimensional smooth manifolds
I am attempting to investigate transformations between two distinct sets of vertices on n-dimensional manifolds with a minimal change in the fundamental shape of the vertices. I will give some ...
- 1,401
6
votes
3
answers
338
views
Large geodesically convex subsets of tori
Let $X=\mathbb T^d=\mathbb R^d/\mathbb Z^d$ and let $E$ be a proper open subset of $X$. We say $E$ is geodesically convex if for any $x,y\in E$ the shortest geodesic connecting $x$ and $y$ lies in $E$....
- 2,085
16
votes
2
answers
1k
views
Behavior of sectional curvature under metric deformations
Metric deformation:
Let $(M,g_0)$ be a Riemannian manifold and consider a (sufficiently smooth) deformation of $g_0$, $$g_t=g_0+th+O(t^2), \quad 0< t<\varepsilon $$ where $h$ is some symmetric (...
- 5,831
3
votes
1
answer
766
views
Conformally-flat
Assume given a smooth manifold $(\mathbb{R}^n, g)$, where the metric is a scaled identity $g = e^{2f}I$.
Is there a way to know if this is always a non-positive (sectional) curvature manifold?
Note ...
- 75
6
votes
1
answer
1k
views
Good Surface,Bad Surface-Surface classification
Maybe this question be very simple, but I don't know why it is hard for me. Thanks for any guide and help.
We say a surface $S$ (2-dimensional metric(compact) Riemannian surface) is good (denote by $...
- 4,748
1
vote
2
answers
359
views
intersection of geodesiques
Let $(M,g)$ be a closed riemannian surface . let $\alpha$ be a simple closed geodesique . does there is exist a simple closed geodesic $\beta$ that intersect alpha at only 1 point p such that $[\...
student
9
votes
6
answers
3k
views
Exponential and Logarithm Mapping on Stiefel Manifold
The Stiefel Manifold is defined as
$$
\mathrm{St}(p,n):= \{ X\in \mathbb{R}^{n\times p} :\ X^T X = I_p \}.
$$
Recall that the tangent space at a point $X\in \mathrm{St}(p,n)$ is given by
$$
T_X{\...
- 969
5
votes
1
answer
280
views
rigidity of eigenvalues of circular ensemble
Given a circular unitary ensemble, with the following joint density:
$p(\theta_1,\ldots, \theta_n) = Z_n \prod_{j < k} |e^{i \theta_j} - e^{i \theta_k}|^2$,
is the following statement true? With ...
- 4,354
10
votes
3
answers
1k
views
volume of compact simple Lie groups under the natural Euclidean embedding
I am looking for a quick reference for the volume formula for all the compact simple Lie groups embedded as matrix groups in the natural way. The one I care most for are the real orthogonal groups. I ...
- 4,354
4
votes
1
answer
1k
views
Length spaces with continuous length functional: is this set Gromov-Hausdorff closed?
As far as I can tell, a major motivation for the study of length spaces is that they arise as Gromov-Hausdorff limits of Riemannian manifolds. Specifically,
A complete connected Riemannian manifold ...
- 3,192
3
votes
1
answer
345
views
extended forms from foliations [closed]
hi,
i have the following question: Let $M$ be a n-dimensional manifold (or riemannian or everything thats nice ...) and let $\mathcal{F}$ be a foliation of $M$ by surfaces. Assume, furthermore, that ...
- 221
3
votes
2
answers
273
views
non commutative elements in the fundamental group of a closed hyperbolic surface
Let $(M,hyp)$ be a closed hyperbolic surface. fix a point $m$ in $M$ and denote by $G=\pi_1(M,m) $.
now let $\alpha$ and $\beta$ in $G$ such that $\alpha$ and $\beta$ does not commute . my first ...
student
15
votes
3
answers
2k
views
Good reference for globally formulated calculus of variations on Riemannian manifolds?
I have been working on a formulation of the calculus of variations on Riemannian manifolds, formulated globally using [pullback] vector bundles and tensor bundles and their induced covariant ...
- 623
10
votes
0
answers
548
views
Killing spinors and symmetric tensor fields.
Hi all,
I have a question of the following form: Let $(M,g)$ be a Riemannian spin manifold which admits a Killing spinor $\sigma$ and let $h:T M \to T M$ be a symmetric, trace-free and divergence-...
- 101