Assertion: complex's structure | /r/HasProperty | integrable
Facets:
value: more generally
type: DEGREE
value: in ordinary complex geometry
type: OTHER
value: if
type: DEGREE
value: to a complex structure
type: OTHER
Source sentences:
text: In ordinary complex geometry, an almost complex structure is integrable to a complex structure if and only if the Lie bracket of two sections of the holomorphic subbundle is another section of the holomorphic subbundle.
text: An almost complex structure that comes from a complex structure is called integrable, and when one wishes to specify a complex structure as opposed to an almost complex structure, one says an integrable complex structure.
text: If a pure spinor that determines a particular complex structure is closed, or more generally if its exterior derivative is equal to the action of a gamma matrix on itself, then the almost complex structure is integrable and so such pure spinors correspond to generalized complex structures.
text: The word "almost" is removed if the structure is integrable.
text: The eigenvalues of an almost complex structure are ±i and the eigenspaces form sub-bundles denoted by T0,1M and T1,0M. The Newlander–Nirenberg theorem shows that an almost complex structure is actually a complex structure precisely when these subbundles are involutive, i.e., closed under the Lie bracket of vector fields, and such an almost complex structure is called integrable.