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Register a new userExtrinsic geometry of the Gromoll-Meyer sphere
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Among a family of 2-parameter left invariant metrics on Sp(2), we determine which have nonnegative sectional curvatures and which are Einstein. On the quotiente $widetilde{N}^{11}=(Sp(2)times S^4)/S^3$, we construct a homogeneous isoparametric foliation with isoparametric hypersurfaces diffeomorphic to Sp(2). Furthermore, on the quotiente $widetilde{N}^{11}/S^3$, we construct a transnormal system with transnormal hypersurfaces diffeomorphic to the Gromoll-Meyer sphere $Sigma^7$. Moreover, the induced metric on each hypersurface has positive Ricci curvature and quasi-positive sectional curvature simultaneously.
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Gromoll and Meyer have represented a certain exotic 7-sphere $Sigma^7$ as a biquotient of the Lie group $G = Sp(2)$. We show for a 2-parameter family of left invariant metrics on $G$ that the induced metric on $Sigma^7$ has strictly positive sectional curvature at all points outside four subvarieties of codimension $geq 1$ which we describe explicitly.
We study the geometry of the tangent bundle equipped with a two-parameter family of Riemannian metrics. After deriving the expression of the Levi-Civita connection, we compute the Riemann curvature tensor and the sectional, Ricci and scalar curvatures. Specializing to the case of space forms, we characterise the metrics giving positive sectional curvature and show that one can always find parameters ensuring positive scalar curvature on the tangent space. Under some curvature conditions, this extends to general base manifolds.
We prove that the group $Aut_1(xi)$ of strict contactomorphisms, also known as quantomorphisms, of the standard tight contact structure $xi$ on $S^3$ is the total space of a fiber bundle $S^1 to Aut_1(xi) to SDiff(S^2)$ over the group of area-preserving $C^infty$-diffeomorphisms of $S^2$, and that it deformation retracts to its finite-dimensional sub-bundle $S^1 to U(2)cup cU(2) to O(3)$, where $U(2)$ is the unitary group and $c$ is complex conjugation.
We study the geometry of infinitely presented groups satisfying the small cancelation condition C(1/8), and define a standard decomposition (called the criss-cross decomposition) for the elements of such groups. We use it to prove the Rapid Decay property for groups with the stronger small cancelation property C(1/10). As a consequence, the Metric Approximation Property holds for the reduced C*-algebra and for the Fourier algebra of such groups. Our method further implies that the kernel of the comparison map between the bounded and the usual group cohomology in degree 2 has a basis of power continuum. The present work can be viewed as a first non-trivial step towards a systematic investigation of direct limits of hyperbolic groups.
We study harmonic sections of a Riemannian vector bundle whose total space is equipped with a 2-parameter family of metrics which includes both the Sasaki and Cheeger-Gromoll metrics. This enables the theory of harmonic unit sections to be extended to bundles with non-zero Euler class.
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