Compact differentiable transformation groups on exotic spheres

Curvature and Symmetry of Milnor Spheres

“…If one chooses the free action by S 3 1 × S 3 2 , it was shown in [GM] that one obtains the principal bundle P 2,−1 and hence the associated sphere bundle P 2,−1 × S 3 1 ×S 3 2 S 3 = P 2,−1 /△ 1,2 S 3 is the exotic sphere M 2,−1 with a submersed metric of nonnegative curvature. As in our case, the action of S 1 ×S 3 3 descends to an action of the associated S 3 bundle M 2,−1 which becomes an effective action by SO(3)SO(2) and which, by [St1,Theorem C], is the id-component of the isometry group of the Gromoll-Meyer metric. Notice that, as in our case, we also get a family of metrics with nonnegative curvature on the Gromoll-Meyer sphere by considering Sp(2) × S 3 1 ×S 3 2 S 3 (r), and as r goes to infinity we obtain the Gromoll-Meyer metric in the limit.…”

“…Proof. To see this, we use the result by E. Straume that the degree of symmetry of any exotic 7-sphere is at most 4, see [St1,Theorem C]. In other words the dimension of any compact Lie group G that acts effectively on an exotic 7-sphere is at most 4.…”

Positively curved manifolds with symmetry

“…Recall that a Riemannian manifold is said to have positive l th Ricci curvature if for any unit vector v, the sum of the l smallest eigenvalues of R(·, v)v, viewed as an endomorphism of (v) ⊥ , is positive. [18] says that an exotic n-sphere which admits a metric with symmetry degree ≥ 3 2 (n + 1) is necessarily given by a Brieskorn variety or equivalently bounds a parallelizable manifold. Notice that for n ≥ 15 the symmetry degree of the manifold in Theorem 1 is necessarily ≥ 3 2 (n + 1).…”

Structure of diffeomorphism groups of non-Archimedean manifolds

“…In the particular case of the local field F = K such subgroups have the following property: the K-linear span sp K (T e G n u,K ) of T e G n u,K is dense in T e G(t, M). This is the important difference from the case of M on X over R or C, because the maximal compact subgroup in G(t, M) in the classical case may be only finite-dimensional for finite-dimensional X over R or C [33]. This also is impossible in the classical case, when M is not a compact complex manifold.…”

“…). This is the important difference from the case of M on X over R or C, because the maximal compact subgroup in G(t, M) in the classical case may be only finite-dimensional for finite-dimensional X over R or C [33]. This also is impossible in the classical case, when M is not a compact complex manifold.…”