  We prove under certain assumptions no-hair theorems for non-canonical
self-gravitating static multiple scalar fields in spherically symmetric
spacetimes. It is shown that the only static, spherically symmetric and
asymptotically flat black hole solutions consist of the Schwarzschild metric
and a constant multi-scalar map. We also prove that there are no static,
horizonless, asymptotically flat, spherically symmetric solutions with static
scalar fields and a regular center. The last theorem shows that the static,
asymptotically flat, spherically symmetric reflecting compact objects with
Neumann boundary conditions can not support a non-trivial self-gravitating
non-canonical (and canonical) multi-scalar map in their exterior spacetime
regions. In order to prove the no-hair theorems we derive a new divergence
identity.
