Tensor bundle

of type $ ( p, q) $ on a differentiable manifold $ M $

The vector bundle $ T ^ {p,q} ( M) $ over $ M $ associated with the bundle of tangent frames and having as standard fibre the space $ T ^ {p,q} ( \mathbf R ^ {n} ) $ of tensors (cf. Tensor on a vector space) of type $ ( p, q) $ on $ \mathbf R ^ {n} $, on which the group $ \mathop{\rm GL} ( n, \mathbf R ) $ acts by the tensor representation. For instance, $ T ^ {1,0} ( M) $ coincides with the tangent bundle $ T M $ over $ M $, while $ T ^ {0,1} ( M) $ coincides with the cotangent bundle $ T ^ {*} M $. In the general case, the tensor bundle is isomorphic to the tensor product of the tangent and cotangent bundles:

$$ T ^ {p,q} ( M) \cong \otimes ^ { p } TM \otimes \otimes ^ { q } T ^ {*} M . $$

Sections of the tensor bundle of type $ ( p, q) $ are called tensor fields of type $ ( p, q) $ and are the basic object of study in differential geometry. For example, a Riemannian structure on $ M $ is a smooth section of the bundle $ T ^ {0,2} ( M) $ the values of which are positive-definite symmetric forms. The smooth sections of the bundle $ T ^ {p,q} ( M) $ form a module $ D ^ {p,q} ( M) $ over the algebra $ F ^ { \infty } ( M) = D ^ {0,0} ( M) $ of smooth functions on $ M $. If $ M $ is a paracompact Hausdorff manifold, then

$$ D ^ {p,q} ( M) \cong \ \otimes ^ { p } D ^ {1} ( M) \otimes \otimes ^ { q } D ^ {1} ( M) ^ {*} , $$

where $ D ^ {1} ( M) = D ^ {1,0} ( M) $ is the module of smooth vector fields, $ D ^ {1} ( M) ^ {*} = D ^ {0,1} ( M) $ is the module of Pfaffian differential forms (cf. also Pfaffian form), and the tensor products are taken over $ F ^ { \infty } ( M) $. In classical differential geometry tensor fields are sometimes simply called tensors on $ M $.

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 $ D ^ {1} ( M) $ of vector fields is often denoted by $ X( M) $, and $ D ^ {1} ( M) ^ {*} $, the space of Pfaffian forms, by $ \Omega ^ {1} ( M) $.

References

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