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

No-hair theorems for non-canonical self-gravitating static multiple scalar fields