On Bär's Conformal Lower Bound for the Spectrum of Generalized Dirac Operators Page: 13
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BOUND ON SPECTRUM OF DIRAC OPERATORS
Since I : TM S - S is onto, we have a canonical bundle isoporphiap
TM @ S N S kier p, implementable via the isomorphism (ker)L -4(
induced by the Clifford multiplication p. By analogy with the spin manifold
case we call ker p the bundle of ntri twistoes and the differential
operator T : C(S) - C(kerp), T = oj o V, the generlid twitor
operator.
LEMMA 2.1. The inverse I : S - (ker)' of th bundle isaoqrplism
(ker}L) - S is gin pointaie byg the formula
1"
(2.2) I(s)= - .sas, s .
(2.3) I()= !1fg, as 9,
(2.4) V = I(D) +T(s), aBE "(s),
(2.5) IVaj = -jZsi+(T(s)' s G"(S).
Proof, We have I(a) = I i a ea;, for some as, a,..., a,, in.. Then
_i e tai = a, since so I(a) = a. Also, ej e'a p e. * a, for any two indice
j and k, since I(a) 1 kerp and Pj @ej "-.-ek @e kerp, for any e 5.
Formula (2.2) follows. Using (2.2), (2.3) is now obvion&
It is clear that for a E C m(S) there is a E C'(S) such that'Va =
I(r) + T(s). Applying p to this equation gives Da = a. Finally, (2.5)
follows immediately from (2.3) and (2.4). 13
Fix now a function f E C(M, R) and assume that a E C(S) is an
eigensection for D with eigenvalue A. Since e/V(e-fs) = -gradf @a + Vs
we have
(2.6) l eV(e-Is) I= JgradfI2Jal2 - (grad.,, )- (, Vgrad,) + (V*, Va).
However, (Vgradfa, a) + (a, Vgradf) = gradf(Isla) = (4f, d(jal2)). Thus
equations (2.5) and (2.6) imply
(2.7)
IgradfltI - (df, d(lelI))+ (VA, Va) = elD- +eTe-f13
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Anghel, Nicolae. On Bär's Conformal Lower Bound for the Spectrum of Generalized Dirac Operators, article, 2000; [Bucharest, Romania]. (https://digital.library.unt.edu/ark:/67531/metadc161690/m1/4/: accessed April 25, 2024), University of North Texas Libraries, UNT Digital Library, https://digital.library.unt.edu; crediting UNT College of Arts and Sciences.