An initial data set for the Einstein constraint equations consists of a Riemannian manifold 
and a symmetric covariant two-tensor 
, representing a metric and second
fundamental form on a spacelike hypersurface. For a cosmological constant 
, one convention defines the constraint
densities 
and 
by
 |
(1)
|
 |
(2)
|
where 
is scalar curvature. The dominant
energy condition is 
.
The choice 
is time symmetry, whereas 
is maximality. For an oriented hypersurface 
in 
, its outward future expansion in this convention is
 |
(3)
|
where 
is the sum of the principal curvatures toward the chosen exterior, rather than their
average as in the mean curvature convention. A
marginally outer trapped surface
has 
.
These constraint and expansion conventions are used in the Penrose
inequality.
