Clifford Bundle - Clifford Bundle of A Riemannian Manifold

Clifford Bundle of A Riemannian Manifold

If M is a Riemannian manifold with metric g, then the Clifford bundle of M is the Clifford bundle generated by the tangent bundle TM. One can also build a Clifford bundle out of the cotangent bundle T*M. The metric induces a natural isomorphism TM = T*M and therefore an isomorphism Cℓ(TM) = Cℓ(T*M).

There is a natural vector bundle isomorphism between the Clifford bundle of M and the exterior bundle of M:

This is an isomorphism of vector bundles not algebra bundles. The isomorphism is induced from the corresponding isomorphism on each fiber. In this way one can think of sections of the Clifford bundle as differential forms on M equipped with Clifford multiplication rather than the wedge product (which is independent of the metric).

The above isomorphism respects the grading in the sense that

$\begin{align} C\ell^0(T^*M) &= \Lambda^{\mathrm{even}}(T^*M)\\ C\ell^1(T^*M) &= \Lambda^{\mathrm{odd}}(T^*M). \end{align}$

Read more about this topic: Clifford Bundle

Famous quotes containing the words clifford, bundle and/or manifold:

“We know all their gods; they ignore ours. What they call our sins are our gods, and what they call their gods, we name otherwise.”
—Natalie Clifford Barney (1876–1972)

““There is Lowell, who’s striving Parnassus to climb
With a whole bale of isms tied together with rhyme,
He might get on alone, spite of brambles and boulders,
But he can’t with that bundle he has on his shoulders,
The top of the hill he will ne’er come nigh reaching
Till he learns the distinction ‘twixt singing and preaching;”
—James Russell Lowell (1819–1891)

“There must be no cessation
Of motion, or of the noise of motion,
The renewal of noise
And manifold continuation....”
—Wallace Stevens (1879–1955)