  We revisit three results due to Morita expressing certain natural integral
cohomology classes on the universal family of Riemann surfaces C_g, coming from
the parallel symplectic form on the universal jacobian, in terms of the
Miller-Morita-Mumford classes e and e_1. Our discussion will be on the level of
the natural 2-forms representing the relevant cohomology classes, and involves
a comparison with other natural 2-forms representing e, e_1 induced by the
Arakelov metric on the relative tangent bundle of C_g over M_g. A secondary
object called a_g occurs, which was discovered and studied by Kawazumi around
2008. We present alternative proofs of Kawazumi's (unpublished) results on the
second variation of a_g on M_g. Also we review some results that were
previously obtained on the invariant a_g, with a focus on its connection with
Faltings's delta-invariant and Hain-Reed's beta-invariant.
