Abstract
The author studies static solutions of the vacuum Einstein equations in D=m+n+2 dimensions which have the following symmetries: the solutions are spherically symmetric in (m+2) dimensions, (or more generally the m-sphere is replace by an arbitrary m-dimensional Einstein space), while the internal space is any arbitrary n-dimensional Einstein space. The global properties of all such solutions are derived by considering the equivalent dimensionally reduced system in (m+2) dimensions, and by using techniques from the theory of dynamical systems after a judicious choice of variables. Apart from the trivial case of the Schwarzschild solution with constant scalar field, all solutions are found to contain naked singularities or else not to be asymptotically flat, as would be expected from the 'no hair' theorems.
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