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We study flat left-invariant pseudo-Riemannian metrics on nilpotent Lie groups. Associated with a flat metric Lie algebra $(\mathfrak{g},\langle\cdot,\cdot\rangle)$ there is a canonical affine action of the corresponding group on the vector space $\mathfrak{g}$, whose linear part is the Levi-Civita representation $L$....
We construct Riemannian metrics of strictly positive sectional curvature on at least 508 pairwise nondiffeomorphic exotic fifteen-spheres, without distinguishing orientations. The construction starts with quaternionic line bundles over the product of the quaternionic projective plane and the four-sphere. Surgery on an...
We study the construction of positively curved Riemannian metrics by non-isometric circle actions. For a family of homotopy eleven-spheres whose Eells--Kuiper invariants form the even subgroup of $\mathbb{Z}/992$, we construct, on each member, a smooth background metric $q$ and three effective circle actions with gener...
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