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An affine surface is said to be an affine Zoll surface if all affine geodesics close smoothly. It is said to be an affine almost Zoll surface if thru any point, every affine geodesic but one closes smoothly (the exceptional geodesic is said to be alienated as it does not return). We exhibit an affine structure on the cylinder which is almost Zoll. This structure is geodesically complete, affine Killing complete, and affine symmetric.
If $mathcal{M}=(M, abla)$ is an affine surface, let $mathcal{Q}(mathcal{M}):=ker(mathcal{H}+frac1{m-1}rho_s)$ be the space of solutions to the quasi-Einstein equation for the crucial eigenvalue. Let $tilde{mathcal{M}}=(M,tilde abla)$ be another affin
We prove an almost splitting theorem for the warped product space with warped function $f(r)=coshleft(rsqrt{frac{lambda}{n-2}}right)$.
We show that any closed spin manifold not diffeomorphic to the two-sphere admits a sequence of volume-one-Riemannian metrics for which the smallest non-zero Dirac eigenvalue tends to zero. As an application, we compare the Dirac spectrum with the conformal volume.
We investigate the geometry of almost Robinson manifolds, Lorentzian analogues of Hermitian manifolds, defined by Nurowski and Trautman as Lorentzian manifolds of even dimension equipped with a totally null complex distribution of maximal rank. Assoc
We find a new class of invariant metrics existing on the tangent bundle of any given almost-Hermitian manifold. We focus here on the case of Riemannian surfaces, which yield new examples of Kahlerian Ricci-flat manifolds in four real dimensions.