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Register a new userThe geometry of generalised Cheeger-Gromoll metrics
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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.
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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.
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.
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 show for a certain class of operators $A$ and holomorphic functions $f$ that the functional calculus $Amapsto f(A)$ is holomorphic. Using this result we are able to prove that fractional Laplacians $(1+Delta^g)^p$ depend real analytically on the metric $g$ in suitable Sobolev topologies. As an application we obtain local well-posedness of the geodesic equation for fractional Sobolev metrics on the space of all Riemannian metrics.
The various scalar curvatures on an almost Hermitian manifold are studied, in particular with respect to conformal variations. We show several integrability theorems, which state that two of these can only agree in the Kahler case. Our main question is the existence of almost Kahler metrics with conformally constant Chern scalar curvature. This problem is completely solved for ruled manifolds and in a complementary case where methods from the Chern-Yamabe problem are adapted to the non-integrable case. Also a moment map interpretation of the problem is given, leading to a Futaki invariant and the usual picture from geometric invariant theory.
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