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In geometry, a hypersurface is a generalization of the concepts of hyperplane, plane curve, and surface. A hypersurface is a manifold or an algebraic variety of dimension n − 1 , which is embedded in an ambient space of dimension n , generally a Euclidean space , an affine space or a projective space . [ 1 ]
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Furthermore, this compact manifold with boundary, which is known as the Milnor fiber (of the isolated singular point of at the origin), is diffeomorphic to the intersection of the closed (+)-ball (bounded by the small (+)-sphere) with the (non-singular) hypersurface where = and is any sufficiently small non-zero complex number.
Its affirmative hand, more general than the Chern's conjecture for hypersurfaces, sometimes also referred to as the Chern's conjecture and is still, as of 2018, unanswered even with M as a hypersurface (Chern proposed this special case to the Shing-Tung Yau's open problems' list in differential geometry in 1982):
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As such, the -sphere is the setting for -dimensional spherical geometry. Considered extrinsically, as a hypersurface embedded in ( n + 1 ) {\displaystyle (n+1)} -dimensional Euclidean space , an n {\displaystyle n} -sphere is the locus of points at equal distance (the radius ) from a given center point.
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