A manifold with a “Lie structure at infinity” is a non-compact manifold M0 whose geometry is described by a compactification to a manifold with corners M and a Lie algebra V of vector fields on M subject to constraints only on M rM0. This definition recovers several classes of non-compact manifolds that were studied before: manifolds with cylindrical ends, manifolds that are Euclidean at infini...

The geography and botany of smooth/symplectic 4-manifolds with cyclic fundamental group are addressed. For all the possible lattice points which correspond to non-spin manifolds of negative signature and a given homeomorphism type, an irreducible symplectic manifold and an infinite family of pairwise non-diffeomorphic non-symplectic irreducible manifolds are manufactured. In the same fashion, a...

The relationship between monotonicity and accretivity on Riemannian manifolds is studied in this paper and both concepts are proved to be equivalent in Hadamard manifolds. As a consequence an iterative method is obtained for approximating singularities of Lipschitz continuous, strongly monotone mappings. We also establish the equivalence between the strong convexity of convex functions and the ...

We develop a construction of Engel structures on 4– manifolds based on decompositions of manifolds into round handles. This allows us to show that all parallelizable 4–manifolds admit an Engel structure. We also show that, given two Engel manifolds M1, M2 satisfying a certain condition on the characteristic foliations, there is an Engel structure on M1#M2#(S ×S) which is closely related to the ...

The aim of this paper is to determine topological types of torus manifolds which have codimension one extended G-actions. As a result, we show that their topological types are completely determined by their cohomology rings and characteristic classes. Due to this result, we find the counterexample to the cohomological rigidity problem in the category of torus manifolds. Moreover, we find the cl...

We completely characterize real Bott manifolds up to diffeomorphism in terms of three simple matrix operations on square binary matrices obtained from strictly upper triangular matrices by permuting rows and columns simultaneously. We also prove that any graded ring isomorphism between the cohomology rings of real Bott manifolds with Z/2 coefficients is induced by an affine diffeomorphism betwe...

We compute for all orientable irreducible geometric 3-manifolds certain complexity functions that approximate from above Matveev’s natural complexity, known to be equal to the minimal number of tetrahedra in a triangulation. We can show that the upper bounds on Matveev’s complexity implied by our computations are sharp for thousands of manifolds, and we conjecture they are for infinitely many, ...

Rigidity results are obtained for Riemannian d-manifolds with sec > 1 and spherical rank at least d − 2 > 0. Conjecturally, all such manifolds are locally isometric to a round sphere or complex projective space with the (symmetric) Fubini– Study metric. This conjecture is verified in all odd dimensions, for metrics on dspheres when d 6= 6, for Riemannian manifolds satisfying the Rakić duality p...