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12
votes
Accepted
Realizing mapping classes as isometries?
As you suspect, in general, no.
For example, if $M$ is compact, and $\psi:M\to M$ fixes a metric $g$ on $M$, then the closure of $\{\psi^k\ |\ k\in\mathbb{Z}\ \}$ is a compact abelian subgroup of $\ma …
22
votes
Isometric embedding of SO(3) into an euclidean space
About embeddings, I don't know, but there is an isometric immersion of $\mathrm{SO}(3)$ with its bi-invariant metric into $\mathbb{R}^7$.
To see this, consider the natural representation $\rho_3:\mat …
7
votes
Accepted
Homogeneous subsets of the sphere
More generally, if $G$ is a connected compact Lie group acting irreducibly by isometries on a (finite-dimensional) Hilbert space $H$, then each of the orbits of $G$ in the unit sphere $S\subset H$ will …
5
votes
Accepted
Conformal harmonic maps in high dimensions are scaled isometries
This result is well-known in the theory of harmonic morphisms, about which, there is an extensive literature. It is a quite general fact (not depending on the conformally flat case of Euclidean space …
5
votes
Accepted
Does the isometry group determine the Riemannian metric?
I think that you are missing some hypotheses on the action of $G$, otherwise there are trivial counterexamples. For example, if $G\subset\mathrm{Iso}(M,g)$ is the trivial group, then one clearly cann …
13
votes
Isometry group of a left-invariant Riemannian metric on $\mathrm{SU}(2)$
Up to a scalar multiple, it is the standard round sphere, so the group of isometries of $g$ is $\mathrm{O}(4)$. …
19
votes
Accepted
Tweetable way to see Riemannian isometries are harmonic?
Not exactly 'tweetable', but perhaps the identity (1) may help, if all you want to do is avoid the Euler-Lagrange equations. For simplicity, assume that $M^n$ is oriented. (One can write the identit …
16
votes
Accepted
The surjectivity of the exponential map for the isometry group
You'll run into the same problem with the conformal group in this example, because the space of conformal transformations in this example is the same as the space of isometries. …
2
votes
Accepted
Are all symmetries of the Dirichlet functional isometries?
The answer to your question is "yes, every symmetry (in the sense you have specified) is an isometric immersion".
To see why, first note that, if $(M,g)$ is a compact Riemannian $m$-manifold and $h$ …
13
votes
Isometry group of a compact hyperbolic surface
Then the group of orientation preserving isometries of $C$ will be an extension of $\Gamma$ by a $\mathbb{Z}_2$ (because of the hyperelliptic involution). … The orientation-preserving isometries of $C$ are exactly the elements of $\mathrm{PSL}(3,\mathbb{C})$ that preserve the quartic $Q$. …
16
votes
Accepted
Are there some intrinsic invariants of surfaces other than Gaussian curvature?
As others have pointed out, it's not hard to show that any function $F(\kappa_1,\kappa_2)$ that is intrinsic to the surface metric must be a function of $K = \kappa_1\kappa_2$, so that settles what on …