  These are notes for a short course and some talks gave at Departament of
Mathematics and at Departament of Physics of Federal University of Minas
Gerais, based on the author's paper arXiv:1808.09249. Some new information and
results are also presented. Unlike the original work, here we try to give a
more physical emphasis. In this sense, we present obstructions to realize
gravity, modeled by the tetradic Einstein-Hilbert-Palatini (EHP) action
functional, in a general geometric setting. In particular, we show that if
spacetime has dimension $n\geq4$, then the cosmological constant plays no role
in any "concrete geometries" other than Lorentzian. If $n\geq6$, then the
entire EHP theory is trivial, meaning that Lorentzian geometry is (essentially)
the only "concrete geometry" in which gravity (i.e, the EHP action functional)
makes sense. Examples of "concrete geometries" include those locally modeled by
group reductions $H\hookrightarrow O(k;A)$ for some $k$ and some algebra $A$,
so that Riemannian geometry, Hermitian geometry, K\"ahler geometry and
symplectic geometry, as well as Type II geometry, Hitchin's generalized complex
geometry and $G_{2}$-geometry are included. We also study EHP theory in
"abstract geometries", such as graded geometry (and hence supergeometry), and
we show how the obstruction results extend to this context. We construct two
theories naturally associated to EHP, which we call the geometric/algebraic
dual of EHP, and we analyze the effect of the obstructions in these new
theories. Finally, we speculate (and provide evidence for) the existence of a
"universal obstruction condition".
