preprint
Inserted: 15 oct 2023
Year: 2023
Abstract:
It is known that on $\mathrm{RCD}$ spaces one can define a distributional
Ricci tensor ${\bf Ric}$. Here we give a fine description of this object by
showing that it admits the polar decomposition $${\bf Ric}=\omega\,
{\bf
Ric}
$$ for a suitable non-negative measure $
{\bf Ric}
$ and unitary tensor
field $\omega$. The regularity of both the mass measure and of the polar vector
are also described. The representation provided here allows to answer some open
problems about the structure of the Ricci tensor in such singular setting. Our
discussion also covers the case of Hessians of convex functions and, under
suitable assumptions on the base space, of the Sectional curvature operator.