  We show that Belinfante construction of an improved energy-momentum tensor
can be carried over to curved backgrounds, in analogy to the case of flat
spacetime. The results hold irrespective of the background being dynamical or a
fixed, non-backreacting one. It turns out that the analogous would-be canonical
energy-momentum tensor is not covariantly conserved in general, but its
Belinfante "improvement" is. We relate this last tensor with the Hilbert tensor
obtained by functionally derivating the Lagrangian with respect to the metric.
When the background in non-dynamical, we discuss some issues concerning the
Noether conserved currents associated with its Killing symmetries and the role
played by the Belinfante tensor. Next we study extended objects ($p$-branes)
either in a dynamic or in a fixed background, and obtain the Noether identities
associated both with target spacetime and world volume diffeomorphisms. We show
that in field theory as well as with extended objects, the Killing symmetries
of the background become ordinary rigid Noether symmetries of the theory in
this fixed background. With the example of Maxwell theory in Minkowski
spacetime we show in an appendix the role of the Belinfante tensor in the
construction of these symmetries.
