Una manera posible de naturalizar el alma humana

One of the most basic linguistic principles (or parameters, according to some) is used to point to a likely necessary (though, probably, not sufficient) condition of human uniqueness (i.e. being able to speak, communicate complex messages, have religious feelings, indulge in art and humour, and so on) which has been conceptualised either as the fact that we have human souls, in religious contexts, or human minds, in other contexts

Focal Radius, Rigidity, and Lower Curvature Bounds

We show that the focal radius of any submanifold $N$ of positive dimension in a manifold $M$ with sectional curvature greater than or equal to $1$ does not exceed $\frac{\pi }{2}.$ In the case of equality, we show that $N$ is totally geodesic in $M$ and the universal cover of $M$ is isometric to a sphere or a projective space with their standard metrics, provided $N$ is closed. Our results also hold for $k^{th}$--intermediate Ricci curvature, provided the submanifold has dimension $\geq k.$ Thus in a manifold with Ricci curvature $\geq n-1,$ all hypersurfaces have focal radius $\leq \frac{\pi }{2},$ and space forms are the only such manifolds where equality can occur, if the submanifold is closed. To prove these results, we develop a new comparison lemma for Jacobi fields that exploits Wilking's transverse Jacobi equation.Comment: The first part of the paper has been rewritten to simplify the proofs of the comparison theory for Wilking's transverse Jacobi equation. We have also corrected minor typos and reordered some of the material to simplify the readin