  Generating an effective theory of lower-dimensional gravity on a submanifold
within an original higher-dimensional theory can be achieved even if the
reduction space is non-compact. Localisation of gravity on such a
lower-dimensional worldvolume can be interpreted in a number of ways. The first
scenario, Type I, requires a mathematically consistent Kaluza-Klein style
truncation down to a theory in the lower dimension, in which case solutions
purely within that reduced theory exist. However, that situation is not a
genuine localisation of gravity because all such solutions have
higher-dimensional source extensions according to the Kaluza-Klein ansatz.
Also, there is no meaningful notion of Newton's constant for such Type I
constructions.
  Types II and III admit coupling to genuinely localised sources in the
higher-dimensional theory, with corresponding solutions involving full sets of
higher-dimensional modes. Type II puts no specific boundary conditions near the
worldvolume aside from regularity away from sources. In a case where the wave
equation separated in the non-compact space transverse to the worldvolume
admits a normalisable zero mode, the Type III scenario requires boundary
conditions near the worldvolume that permit the inclusion of that zero mode in
mode expansions for gravitational wave fluctuations or potentials. In such a
case, an effective theory of lower-dimensional gravity can emerge at
sufficiently large worldvolume distance scales.
  This taxonomy of brane gravity localisations is developed in detail for
linearised perturbations about a background incorporating the vacuum solution
of Salam-Sezgin theory when embedded into ten-dimensional supergravity with a
hyperbolic non-compact transverse space. Interpretations of the Newton constant
for the corresponding Type III localisation are then analysed.
