Pullback of Differential Forms
A particular important case of the pullback of covariant tensor fields is the pullback of differential forms. If α is a differential k-form, i.e., a section of the exterior bundle ΛkT*N of (fiberwise) alternating k-forms on TN, then the pullback of α is the differential k-form on M defined by the same formula as in the previous section:
for x in M and Xj in TxM.
The pullback of differential forms has two properties which make it extremely useful.
1. It is compatible with the wedge product in the sense that for differential forms α and β on N,
2. It is compatible with the exterior derivative d: if α is a differential form on N then
Read more about this topic: Pullback (differential Geometry)
Famous quotes containing the words differential and/or forms:
“But how is one to make a scientist understand that there is something unalterably deranged about differential calculus, quantum theory, or the obscene and so inanely liturgical ordeals of the precession of the equinoxes.”
—Antonin Artaud (1896–1948)
“The poet’s eye, in a fine frenzy rolling,
Doth glance from heaven to earth, from earth to heaven;
And as imagination bodies forth
The forms of things unknown, the poet’s pen
Turns them to shapes, and gives to airy nothing
A local habitation and a name.”
—William Shakespeare (1564–1616)