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

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.

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.

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.

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.

On the monodromy theorem for isolated hypersurface singularities.

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.

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

There are only six points on the hypersurface where the axis intersects it that form threespheres.

S** is moving so fast with respect to the tachyon source that the hypersurface becomes titled so much that the arrow points into the past below the hypersurface.
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