differential geometry Differential geometry is a Mathematics, mathematical discipline that studies the geometry of smooth shapes and smooth spaces, otherwise known as smooth manifolds. It uses the techniques of Calculus, single variable calculus, vector calculus, lin ...

, a Clifford module bundle, a bundle of Clifford modules or just Clifford module is a

vector bundle In mathematics, a vector bundle is a topological construction that makes precise the idea of a family of vector spaces parameterized by another space X (for example X could be a topological space, a manifold, or an algebraic variety): to eve ...

whose fibers are Clifford modules, the representations of

Clifford algebra In mathematics, a Clifford algebra is an algebra generated by a vector space with a quadratic form, and is a unital associative algebra with the additional structure of a distinguished subspace. As -algebras, they generalize the real number ...

s. The canonical example is a

spinor bundle In differential geometry, given a spin structure on an n-dimensional orientable Riemannian manifold (M, g),\, one defines the spinor bundle to be the complex vector bundle \pi_\colon\to M\, associated to the corresponding principal bundle \pi_\co ...

. In fact, on a Spin manifold, every Clifford module is obtained by twisting the spinor bundle. The notion "Clifford module bundle" should not be confused with a

Clifford bundle In mathematics, a Clifford bundle is an algebra bundle whose fibers have the structure of a Clifford algebra and whose local trivializations respect the algebra structure. There is a natural Clifford bundle associated to any (pseudo) Riemannian man ...

, which is a bundle of Clifford algebras.

Spinor bundles

Given an oriented

Riemannian manifold In differential geometry, a Riemannian manifold is a geometric space on which many geometric notions such as distance, angles, length, volume, and curvature are defined. Euclidean space, the N-sphere, n-sphere, hyperbolic space, and smooth surf ...

''M'' one can ask whether it is possible to construct a bundle of irreducible Clifford modules over ''Cℓ''(''T''*''M''). In fact, such a bundle can be constructed if and only if ''M'' is a spin manifold. Let ''M'' be an ''n''-dimensional spin manifold with spin structure ''F''_Spin(''M'') → ''F''_SO(''M'') on ''M''. Given any ''Cℓ''_''n''R-module ''V'' one can construct the associated

S(M) = F_(M) \times_\sigma V\,

where σ : Spin(''n'') → GL(''V'') is the representation of Spin(''n'') given by left multiplication on ''S''. Such a spinor bundle is said to be ''real'', ''complex'', ''graded'' or ''ungraded'' according to whether on not ''V'' has the corresponding property. Sections of ''S''(''M'') are called

spinor In geometry and physics, spinors (pronounced "spinner" IPA ) are elements of a complex numbers, complex vector space that can be associated with Euclidean space. A spinor transforms linearly when the Euclidean space is subjected to a slight (infi ...

s on ''M''. Given a spinor bundle ''S''(''M'') there is a natural bundle map :

C\ell(T^*M) \otimes S(M) \to S(M)

which is given by left multiplication on each fiber. The spinor bundle ''S''(''M'') is therefore a bundle of Clifford modules over ''Cℓ''(''T''*''M'').

Notes

References

* * Riemannian geometry Structures on manifolds Clifford algebras Vector bundles {{Riemannian-geometry-stub

Spinor bundles

See also

Notes

References