Tensor bundle

of type on a differentiable manifold

The vector bundle over associated with the bundle of tangent frames and having as standard fibre the space of tensors (cf. Tensor on a vector space) of type on , on which the group acts by the tensor representation. For instance, coincides with the tangent bundle over , while coincides with the cotangent bundle . In the general case, the tensor bundle is isomorphic to the tensor product of the tangent and cotangent bundles:

Sections of the tensor bundle of type are called tensor fields of type and are the basic object of study in differential geometry. For example, a Riemannian structure on is a smooth section of the bundle the values of which are positive-definite symmetric forms. The smooth sections of the bundle form a module over the algebra of smooth functions on . If is a paracompact Hausdorff manifold, then

where is the module of smooth vector fields, is the module of Pfaffian differential forms (cf. also Pfaffian form), and the tensor products are taken over . In classical differential geometry tensor fields are sometimes simply called tensors on .

References

[1]	S. Kobayashi, K. Nomizu, "Foundations of differential geometry" , 1 , Interscience (1963)
[2]	S. Helgason, "Differential geometry, Lie groups, and symmetric spaces" , Acad. Press (1978)

Comments

The space of vector fields is often denoted by , and , the space of Pfaffian forms, by .

References

[a1]	W. Klingenberg, "Riemannian geometry" , de Gruyter (1982) (Translated from German)