Partition of unity

A partition of unity is a locally finite family of smooth functions whose sum is one and whose supports fit within sets of a given open cover.

Support of a function: The support of a function is the closure of the set on which it is nonzero. Each function’s support must lie inside a designated set of the cover.
Riemannian metric: A Riemannian metric is a smoothly varying positive-definite inner product on each tangent space of a manifold. Local metrics can be averaged with partition functions to produce a global metric.
Smooth function: A smooth function has derivatives of every order throughout its domain. The functions in a smooth partition must retain differentiability of every order.
Partition of unity subordinate to a cover: A subordinate partition of unity has each function’s support contained in a corresponding set of a specified cover. This is the precise cover-relative form used in geometric constructions.
Locally finite family: A family of sets is locally finite if every point has a neighborhood meeting only finitely many members. Local finiteness ensures the sum defining the partition is locally a finite sum.
Differential form: A differential form is a smooth field of alternating multilinear functions on tangent spaces. Local differential forms can be combined into a global form using weighted sums.
Subordinate cover: A cover is subordinate to another cover when each of its sets lies inside a member of the latter. Partitions are built using refinements whose pieces fit within the original cover.
Finite partition of unity: A finite partition of unity is a finite family of functions whose pointwise sum is one. It is a special case where local finiteness is automatic.
Smooth bump function: A smooth bump function is a smooth function that vanishes outside a compact region and is nonzero somewhere. Bump functions provide the localized smooth pieces used to build partitions.
Sheaf: A sheaf assigns local data to open sets and specifies how compatible data restrict and glue. Partitions help glue local sections of sheaves of smooth modules.

Linked from 11 pages