# Tensor bundle

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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)

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