Definitions
from The Century Dictionary.
 noun In space of ndimensions, S_{n}, the aggregate of points whose coördinates satisfy a rational homogeneous equation with integral coefficients.
from Wiktionary, Creative Commons Attribution/ShareAlike License.
 noun mathematics A n.dimensional surface in a space (often a
Euclidean space ) of dimension n.+1
Etymologies
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Examples

The Weyl curvature is zero on both the future boundary and past boundary, hence the Big Bang is still welldefined in the cyclic model as the unique hypersurface on which the Weyl curvature vanishes.
Archive 20090501 Gordon McCabe 2009

Found via Exporting the past into the future, or, "The Possibility Jelly lives on the hypersurface of the present" which in turn came via a tweet.
The locationaware future is here (and why cities suck but are good) Mia 2009

Found via Exporting the past into the future, or, "The Possibility Jelly lives on the hypersurface of the present" which in turn came via a tweet.
Archive 20090201 Mia 2009

The far future of our universe is therefore representable by de Sitter spacetime, and in this type of spacetime the future timelike boundary is a spacelike hypersurface.
Archive 20090501 Gordon McCabe 2009

Now, in an open or flat FriedmannRobertsonWalker cosmological model with no cosmological constant, whilst the past 'BigBang' singularity corresponds to a spacelike hypersurface in the boundary of the conformal compactification, the future timelike infinity corresponds to a single point.
Archive 20090501 Gordon McCabe 2009

Given an initial condition on a timelike hypersurface as a mathematical model for the universe, we can determine in which direction a dynamics (supposed here to be deterministic, whether applied to a quantum or to a classical state) causes the entropy to increase or to decrease.
Arrow of Time FAQ Sean 2007

Oh and BTW, the mind is a maximum hypersurface and thought a trajectory on it and the amygdala and hippocampus are Hopf maps of it.

Oh and BTW, the mind is a maximum hypersurface and thought a trajectory on it and the amygdala and hippocampus are Hopf maps of it.

On the monodromy theorem for isolated hypersurface singularities.

The shock waves the hypersphere gave off were only "sound" along one surface (hypersurface?) of the concentrically expanding ripples of pressure.
Orphans of Chaos 2005
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