# 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) |

**How to Cite This Entry:**

Tensor bundle. A.L. Onishchik (originator),

*Encyclopedia of Mathematics.*URL: http://www.encyclopediaofmath.org/index.php?title=Tensor_bundle&oldid=17758