This article introduces the concept of geodesic mappings of manifolds with idempotent pseudo-connections. The basic equations of canonical geodesic mappings of manifolds with completely idempotent pseudo-connectivity and semi-Riemannian manifolds with a degenerate metric are obtained. It is proved that semi-Riemannian manifolds admitting concircular fields admit completely canonical geodesic mappings and form a closed class with respect to these mappings.
<abstract><p>We derive a matrix version of Li & Yau–type estimates for positive solutions of semilinear heat equations on Riemannian manifolds with nonnegative sectional curvatures and parallel Ricci tensor, similarly to what R. Hamilton did in <sup>[<xref ref-type="bibr" rid="b5">5</xref>]</sup> for the standard heat equation. We then apply these estimates to obtain some Harnack–type inequalities, which give local bounds on the solutions in terms of the geometric quantities involved.</p></abstract>
AbstractWe prove the macroscopic cousins of three conjectures: (1) a conjectural bound of the simplicial volume of a Riemannian manifold in the presence of a lower scalar curvature bound, (2) the conjecture that rationally essential manifolds do not admit metrics of positive scalar curvature, (3) a conjectural bound of $$\ell ^2$$
ℓ
2
-Betti numbers of aspherical Riemannian manifolds in the presence of a lower scalar curvature bound. The macroscopic cousin is the statement one obtains by replacing a lower scalar curvature bound by an upper bound on the volumes of 1-balls in the universal cover.
It is shown that a complete Riemannian manifold with boundary is uniquely determined, up to isometry, by its distance difference representation on the boundary. Unlike previously known results, no restrictions on the boundary are imposed.