Bull. Math. Soc. Sci. Math. Roumanie Tome 52(100) No. 3, 2009, 383–386
On the twisted Dirac operator by Adriana Turtoi To Professor S. Ianu¸s on the occasion of his 70th Birthday
Abstract In [3], [2], lower bounds are given for the eigenvalues of the so called “submanifold twisted Dirac operator” DH and their limiting cases are discussed when the mean curvature H 6= 0. We showed in [6] that Hijazi-Zhang’s theorem is true as well when the considered spin manifold is minimal, and therefore H = 0.
Key Words: Dirac operator, spin structure, submanifold, twisted Dirac operator 2000 Mathematics Subject Classification: Primary 53C27, Secondary 53C80. 1
On the twisted Dirac operator
We review our results regarding the lower bounds estimation for the eigenvalues of submanifold twisted Dirac operator when H = 0. f, G) be a Riemannian We will generally preserve the notations and the definitions of [2]. Let (M (m + n)− dimensional spin manifold and let M be an immersed oriented m− dimensional submanifold f with the induced Riemannian structure g = G|M . Assume that (M, g) is spin. Let N M be the in M f. A spin structure also exists on N M [5]. normal bundle of M in M f be the spinor bundles over M, N M and M f respectively. The restricted Let ΣM, ΣN and ΣM f|M can be identified with Σ =: ΣM ⊗ ΣN if mn is even, [2]. If m and n are spinor bundle S := ΣM both odd, one has to take Σ as the direct sum of two copies of that bundle. f|M such that (e1 , ..., Let (e1 , ..., em , ν1 , ..., νn ) be a positively oriented local orthonormal basis of T M em ) (resp. (ν1 , ..., νn )) is a positively oriented local orthonormal basis of T M (respectively of N M ). e denotes the Levi-Civita connection of (M f, G), then for all X ∈ Γ(T M ), for all A ∈ Γ(T N ) and If ∇ i = 1, ..., m, the Gauss formula may be written, as
where
e i (X + A) = ∇i (X + A) + h(ei , X) − h∗ (ei , A) ∇
(1.1)
N ∇i (X + A) = ∇M ei X + ∇ei A,
(1.2)
h∗ (ei ,.) is the transpose of second fundamental form h viewed as a linear map from T M to N M, and e i stands for ∇ ee . ∇ i
Adriana Turtoi
384
e and ∇ the induced spinorial covariant derivative on Γ(S). Therefore, on Γ(S) we Denote also by ∇ have: e = ∇
(
(∇ΣM ⊗ Id + Id ⊗ ∇ΣN ) ⊕ (∇ΣM ⊗ Id + Id ⊗ ∇ΣN ), ∇
ΣM
ΣN
⊗ Id + Id ⊗ ∇
if m, n odd otherwise
,
(1.3)
The spinorial Gauss formula is [1] m X e i ϕ = ∇i ϕ + 1 ej .hij .ϕ, (∀)ϕ ∈ Γ(S). ∇ 2 j=1
The following submanifold Dirac’s operators, [2], may be introduced: e= D
m X i=1
e i, D = ei · ∇
where we have denoted by H =:
Pm
i=1
m X
ei .∇i , DH = (−1)n ω⊥ · D +
i=1
1 H.ω⊥ , 2
(1.4)
h(ei , ei ) the mean curvature vector field and where:
ω⊥ =
(
ωn ,
for n even,
iωn ,
for n odd,
ωn denoting the complex volume form: ωn = i[
n+1 ] 2
ν1 ...νn .
In both cases ( ω⊥ )2 = (−1)n . f, G), therefore we have H = 0 and for all We consider that (M, g) is a minimal submanifold of (M ϕ ∈ Γ(S) DH ϕ = (−1)n ω⊥ Dϕ,
(1.5)
Recall that there exists a Hermitian inner product on Γ(S), denoted by < ·, · >, such that Clifford f|M is skew-symmetric. In the following, we write (·, ·) = Re(< ·, · >). multiplication by a vector of T M N For any spinor field ϕ ∈ Γ(S), let Rϕ : Mϕ → R be N Rϕ := 2
m X
(ei .ej .Id ⊗ ReNi ej ϕ,
i,j=1
ϕ ) | ϕ |2
(1.6)
and Mϕ := {x ∈ M | ϕ(x) 6= 0}, where ReNi ej be the spinorial normal curvature tensor [2]. Theorem 1.1. Let (M, g) be a compact m − dimensional immersed, minimal spin submanifold in the f, G). Then, denoting the scalar curvature of (M, g) by R, we have spin manifold (M λ2 ≥ inf
Mϕ
1 N (R0 + Rϕ ), 4
(1.7)
where R0 = inf M R and λ is an eigenvalue of the twisted Dirac operator DH . Proof: Using (1.3), the Schr¨ odinger-Lichnerowicz formula for the twisted Dirac operator DH becomes for all spinor ϕ ∈ Γ(S) 2 DH ϕ=
m m X 1 X R (Id ⊗ Id)ϕ + ei ej Id ⊗ ReNi ej ϕ − ∇i ∇i ϕ 4 2 i,j=1 i=1
(1.8)
On the twisted Dirac operator
385
and, therefore, we obtain N R + Rϕ | ϕ |2 + | ∇ϕ |2 . (1.9) 4 Suppose that ϕ ∈ Γ(S) is a non-zero eigenvalue spinor of the twisted Dirac operator, so that DH ϕ = λϕ. Therefore, (1.9) gives N R + Rϕ (λ2 − ) | ϕ |2 =| ∇ϕ |2 . 4 Assuming by absurd that N R + Rϕ 0 i) the following inequality holds λ2 ≥
N R + Rϕ m inf . m − 1 Mϕ 4
(1.10)
q m N ) then the following equations are satisfied: inf Mϕ (R + Rϕ ii) If λ = ± 12 m−1 ∇X ϕ +
1 2m
r
m N )X.ω .ϕ = 0 (R + Rϕ ⊥ m−1
(1.11)
∇X ϕ −
1 2m
r
m N )X.ω .ϕ = 0 (R + Rϕ ⊥ m−1
(1.12)
for all X ∈ Γ(T M ).
Proof: Let ϕ be an eigenspinor for the twisted Dirac operator DH , so that DH ϕ = λϕ. We consider the modified connection λ
n
∇im = ∇i + (−1)[ 2 ]
λ ei .ω⊥ , (∀)i = 1, ..., m. m
(1.13)
We can easily compute λ
| ∇ m ϕ |2 =
m X λ λ λ2 | ϕ |2 . (∇im ϕ, ∇im ϕ) =| ∇ϕ |2 − m i=1
(1.14)
The equality (1.14) is a consequence of the fact that the Clifford multiplication with ei , i = 1, ..., m, and γj , j = 1, ..., m is orthogonal and so m X n n ( ∇i ϕ, ei ω⊥ ϕ) = (−1)[ 2 ]+1 (DH ϕ, ϕ) = (−1)[ 2 ]+1 λ | ϕ |2 . i=1
On the other hand, using Schr¨ odinger- Lichnerowicz formula (1.8), (1.14) and (1.9) we obtain ((DH −
N λ R + Rϕ λ 2 m−1 2 λ ] | ϕ |2 + | ∇ m ϕ | 2 ) ϕ, ϕ) = [ − m 4 m2
Adriana Turtoi
386 and by a direct computation ((DH −
(m − 1)2 2 λ 2 λ | ϕ |2 . ) ϕ, ϕ) = m m2
Comparing the last two relations, we obtain [
N λ R + Rϕ m−1 2 − λ ] | ϕ |2 + | ∇ m ϕ |2 = 0. 4 m
(1.15)
Because ϕ 6= 0, this q equality (1.15) implies (1.10). m N ), the equality (1.15) implies that R + RN is constant, M = M ii) If λ = ± 12 m−1 inf Mϕ (R + Rϕ ϕ ϕ λ
λ
and | ∇ m ϕ |2 = 0. So, ∇ m ϕ = 0, and the definition (1.13) implies respectively the equations (1.11), (1.12). Compare theorem 1.2. with Hijazi-Zhang’s theorem [3], [2], which is proved when H 6= 0, note that this is also true for H = 0, as shown in formula (1.10). As an example, in [6], we showed that every spin K¨ ahler manifold is a totally geodesic submanifold of its twistor space and we studied its twisted Killing spinors.
References ¨r, Extrinsec Bound for Eigenvalues of the Dirac Operator, Ann. Glob. Anal. Geom. 16, 1988, [1] C. Ba pp. 573-596. [2] Nicolas Ginoux, Bertrand Morel, On eigenvalue estimates for the submanifold Dirac operator, Int. J. Math. 13 (2002), 533–548. [3] O. Hijazi, X. Zhang, Lower bounds for the Eigenvalues of Dirac Operator II. The Submanifold Dirac Operator, Ann. Glob. Anal. Geom, 20 (2001), 163–181. [4] S. Kobayashy, K. Nomizu, Foundations of Differential Geometry, I, Interscience Publishers, New York, London, 1963. [5] J. Milnor, Remarks concerning spin manifolds, Princeton Univ. Press (1965), 55-62. [6] A. Turtoi, Twisted Dirac Operators on minimal submanifolds, Rendiconti del Circolo Matematico di Palermo, 55,2, 2006, p. 192-202.
Received: 23.03.2009.
University of Bucharest Faculty of Mathematics 14 Academiei str. 70109 Bucharest, Romania E-mail:
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