  Global properties of static, spherically symmetric configurations of scalar
fields of sigma-model type with arbitrary potentials are studied in $D$
dimensions, including space-times containing multiple internal factor spaces.
The latter are assumed to be Einstein spaces, not necessarily Ricci-flat, and
the potential $V$ includes contributions from their curvatures. The following
results generalize those known in four dimensions: (A) a no-hair theorem: in
case $V\geq 0$, an asymptotically flat black hole cannot have varying scalar
fields or moduli fields outside the event horizon; (B) nonexistence of
particlelike solutions in models with $V\geq 0$; (C) nonexistence of wormholes
under very general conditions; (D) a restriction on possible global causal
structures (represented by Carter-Penrose diagrams). The list of structures in
all models under consideration is the same as is known for vacuum with a
cosmological constant in general relativity: Minkowski (or AdS), Schwarzschild,
de Sitter and Schwarzschild--de Sitter, and horizons which bound a static
region are always simple. The results are applicable to a wide range of
Kaluza-Klein, supergravity and stringy models with multiple dilaton and moduli
fields.
