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In mathematics, a Riemannian manifold is said to be flat if its Riemann curvature tensor is everywhere zero. Intuitively, a flat manifold is one that "locally looks like" Euclidean space in terms of distances and angles, e.g. the interior angles of a triangle add up to 180°. The universal cover of a complete flat manifold
In particular, if R is an integral domain, / is flat only if equals R or is the zero ideal. Over an integral domain, a flat module is torsion free . Thus a module that contains nonzero torsion elements is not flat.
Therefore flat modules, and in particular free and projective modules, are torsion-free, but the converse need not be true. An example of a torsion-free module that is not flat is the ideal ( x , y ) of the polynomial ring k [ x , y ] over a field k , interpreted as a module over k [ x , y ].
The null sign (∅) is often used in mathematics for denoting the empty set. The same letter in linguistics represents zero , the lack of an element. It is commonly used in phonology , morphology , and syntax .
A map of rings is called flat if it is a homomorphism that makes B a flat A-module. A morphism of schemes is called faithfully flat if it is both surjective and flat. [2] Two basic intuitions regarding flat morphisms are: flatness is a generic property; and; the failure of flatness occurs on the jumping set of the morphism.
A manifold is asymptotically simple if it admits a conformal compactification ~ such that every null geodesic in has future and past endpoints on the boundary of ~.. Since the latter excludes black holes, one defines a weakly asymptotically simple manifold as a manifold with an open set isometric to a neighbourhood of the boundary of ~, where ~ is the conformal compactification of some ...
An intersection of flats is either a flat or the empty set.. If each line from one flat is parallel to some line from another flat, then these two flats are parallel.Two parallel flats of the same dimension either coincide or do not intersect; they can be described by two systems of linear equations which differ only in their right-hand sides.
In mathematics, Hilbert's Nullstellensatz (German for "theorem of zeros", or more literally, "zero-locus-theorem") is a theorem that establishes a fundamental relationship between geometry and algebra. This relationship is the basis of algebraic geometry. It relates algebraic sets to ideals in polynomial rings over algebraically closed fields.