Tagged Questions

106 views

Riemannian metric and Volume form for $SE(n)$ and/or $E(n)$

I wonder what happens when you construct the Tiling spaces considering the natural action of $SE(n)$ or $E(n)$ rather than $\mathbb R^n$. In order to do that, I need to understand both the (left ...
273 views

Bi invariant Riemannian metric on a Lie Group

I'm trying to find an example of a Lie group $G$ which admits a bi-invariant Riemannian metric, and which has a closed subgroup $H$ such that the manifold $G/H$ does not admit a $G$-invariant ...
111 views

Manifold with a quasi-positive curvature

As far as I know, in a simply connected compact manifold, still there exists no well-known obstruction for a manifold with a quasi-positive curvature to be a manifold with positive curvature. But ...
262 views

Isometry group of pseudo Riemannian manifold always a Lie group? (Myers-Steenrod)

Myers-Steenrod states that the isometry group of a Riemannian manifold is a Lie group. Is that also true for pseudo Riemannian manifolds? I didn't find anything related to that. Cheers
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Invariants of a $GL(3,\mathbb{R})$ action

I'm trying to understand the standard $GL(3,\mathbb{R})$ action on the 15-dimensional space of possible values for the derivative of the Riemann curvature tensor of a 3-dimensional manifold $M$ at a ...
620 views

Is every topological (resp. Lie-) group the isometrygroup of a metric space (resp. Riemannian manifold)?

The isometry group of a metric space is a topological group (with the compact open topology). The isometry group of a Riemann Manifold is a Liegroup. (Thm. of Steenrod-Myers) So, is every topological ...
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Isometric embedding of a compact Lie Group in $M(n,\mathbb{C})$

Greetings, Let $G$ be a compact Lie group with a bi-invariant inner product $h$ on it. Can one embedd $G$ in $M(n,\mathbb{C})$ isometrically for some $n \in \mathbb{N}$. By isometrically I mean that ...
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“Homogeneity” of the Hopf fibration $S^7\to S^{15}\to S^8$ [closed]

My question has to do with an apparent contradiction I get regarding the Hopf fibration $S^7\to S^{15}\to S^8$. Namely, the two following statements cannot be true at the same time (but I do not see ...
181 views

Sobolev Norm of distance function on Riemannian manifold

Suppose $M$ is a Riemannian manifold with distance function $d:M\times M \rightarrow [0,\infty)$. If it helps let $M$ be a Lie group with finite Haar measure $\mu$ and left invariant metric (like ...
731 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. ...
107 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 ...
497 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 ...
489 views

Ricci curvature of the symplectic group

Is the Ricci curvature of the compact symplectic group $Sp(n)$ bounded below by $cn$ for some constant $c > 0$ independent of $n$? For $O(n)$ and $U(n)$ I know many references which state such a ...
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Conjugate points in Lie groups with left-invariant metrics

For any Lie group $G$ there exist many left-invariant Riemannian metrics, namely, one just takes any inner product on the tangent space at the identity $T_eG$ and then left translate it to the other ...
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average Riemannian distance between Identiity and a random point in SO(n) or SU(n)

I can compute the even moments of the Riemannian distance $d(Id, U)$ between the identity element and a uniformly chosen point on say $SU(n)$. But the odd moments elude me. Basically one needs to ...
2k views

$G_2$ and Geometry

In a recent question Deane Yang mentioned the beautiful Riemannian geometry that comes up when looking at $G_2$. I am wondering if people could expand on the geometry related to the exceptional Lie ...
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What are the computationally useful ways of thinking about Killing fields?

One definition of the Killing field is as those vector fields along which the Lie Derivative of the metric vanishes. But for very many calculation purposes the useful way to think of them when dealing ...
607 views

The orthogonal group of a riemannian metric

Let the inner product of the vectors X and Y on a given four dimensional manifold (EDIT: make this R4) be defined as (X*Y) = gikXiYk; using the summation convention for repeated indicies. Let A be a ...
498 views

Action of the group of isometries on a manifold

Hi guys, I am able to prove that any symmetric manifold is complete (Consider a local geodesic and use the symmetry to flip it, effectively doubling the length of the geodesic, ad infinitum). I want ...
Say a manifold M has 3 vector fields S,T and R whose Lie brackets satisfy the equations $[S,T]=R$, $[R,S]=T$ and $[T,R]=S$ Then I suppose the following properties hold for M, There exists a ...
Let $G$ be a compact lie group and $H$ a closed subgroup and hence think of $G/H$ as a homogeneous space. Then how are the Killing fields on $G/H$ the projection of the right-invariant vector fields ...