  Positive curvature and bosons
  Compact positive curvature Riemannian manifolds M with symmetry group G allow
Conner-Kobayashi reductions M to N, where N is the fixed point set of the
symmetry G. The set N is a union of smaller-dimensional totally geodesic
positive curvature manifolds each with even co-dimension. By Berger, N is not
empty. By Lefschetz, M and N have the same Euler characteristic. By Frankel,
the sum of dimension of any two components in N is smaller than the dimension
of M. Reverting the process N to M allows to build up positive curvature
manifolds from smaller ones using division algebras and the geodesic flow. From
dimension 6 to 24, only four exceptional manifolds have appeared so far, some
of them being flag manifolds and related to the special unitary group in three
dimensions. We can now draw a periodic system of elements of the known
even-dimensional positive curvature manifolds and observe that the list of
even-dimensional known positive curvature manifolds has an affinity with the
list of known force carriers in physics. Positive mass of the boson matches up
with the existence of of two linearly independent harmonic k-forms on the
manifold. This motivates to compute more quantities of the exceptional positive
curvature manifolds like the lowest non-zero eigenvalues of the Hodge Laplacian
L=d d*+d* d or properties of the pairs (u,v) of harmonic 2,4 or 8 forms in the
positive mass case.
