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Eigenvalues Estimates For The Dirac Operator In Terms Of Codazzi Tensors

We prove a lower bound for the first eigenvalue of the Dirac operator on a compact Riemannian spin manifold depending on the scalar curvature as well as a chosen Codazzi tensor. The inequality generalizes the classical estimate from [2].


Introduction
The first author proved in [2] that the smallest eigenvalue λ 1 of the Dirac operator D of a compact Riemannian spin manifold (M n , g) satisfies where S min denotes the minimum of the scalar curvature. The limiting case of (1) occurs if and only if (M n , g) admits a nontrivial spinor field ψ 1 satisfying where X is an arbitrary vector field on M n and the dot "·" indicates Clifford multiplication [3]. Improvements of this estimate do typically depend on additional geometric structures on the considered manifold (M n , g) [4], [8], [9]. The aim of this paper is to show that inequality (1) can be improved in case that a Codazzi tensor exists.
A symmetric (0, 2)-tensor field β on (M n , g) is called a nondegenerate Codazzi tensor [1] if β is nondegenerate at all points of M n and satisfies (∇ X β)(Y, Z) = (∇ Y β)(X, Z) for all vector fields X, Y, Z. We identify β with the induced (1, 1)-tensor β via β(X, Y ) = g(X, β(Y )). Let (E 1 , . . . , E n ) be a local orthonormal frame field on (M n , g). Then the spin derivative ∇ and the Dirac operator D, acting on sections ψ ∈ Γ(Σ(M n )) of the spinor bundle Σ(M n ) over (M n , g), are locally expressed as [3] respectively. Moreover, we define the β-twist D β of the Dirac operator D by Theorem 1.1. Let (M n , g) be an n-dimensional closed Riemannian spin manifold and consider a nondegenerate Codazzi tensor β. Denote by g the metric induced by β via g(X, Y ) = g(β(X), β(Y )). Let λ 1 ∈ R and λ 1 ∈ R be the smallest eigenvalue of the Dirac operators D and D, respectively. We assume that both λ 1 and λ 1 are nonzero. Then we have △F := −(div • grad)(F ), and p, q : M n −→ R are bounded real-valued functions satisfying that solve the system of two linear equations The limiting case of (2) occurs if and only if there exists a spinor field ψ 1 on (M n , g) such that hold for some nonzero constantsλ 1 = 0, λ 1 = 0 ∈ R and for all vector fields X. In the limiting case, the parameter c = λ 1 /λ 1 is the ratio of the two eigenvalues.
If β = g = I is the identity map and p + q = −1/n, then (2) reduces to the inequality (1). If the eigenvalues of β = I are constant, but not equal, the solutions p, q of the linear system are constant, too, Consequently, we obtain a family of inequalities depending on a parameter c = 0 linking λ 2 1 , λ 2 1 and S min , The optimal parameter c is a solution of a quadratic equation, we omit the corresponding formulas. A universal though not optimal value for the parameter c is In this case we have p = 0 and q = −1/|β −1 | 2 . This particular inequality generalizes (1): If trβ −1 = 0, the functions p and q do not depend on the parameter c, i. e. we obtain a unique inequality. We will formulate the result separately.
Theorem 1.2. Let (M n , g) be an n-dimensional closed Riemannian spin manifold and consider a nondegenerate Codazzi tensor such that tr(β −1 ) = 0 vanishes identically. Let λ 1 and λ 1 be the smallest eigenvalue of D and D, respectively. We assume that both λ 1 and λ 1 are nonzero. Then, in the notations of Theorem 1.1, we have where the real-valued function F : M n −→ R is defined by The limiting case of (8) occurs if and only if there exists a spinor field ψ 1 on (M n , g) such that (10) holds for some constants λ 1 = 0, λ 1 = 0 ∈ R and for all vector fields X.
Let us discuss the 2-dimensional case in detail. Suppose that β is traceless with eigenvalues a , −a. Then we obtain In particular, the formula of the latter Theorem simplifies: be a 2-dimensional closed Riemannian spin manifold with a nondegenerate traceless Codazzi tensor. Denote by ±a its eigenvalues. Then we have We apply the Corollary to minimal surfaces M 2 ⊂ X 3 (κ) in a 3-dimensional space of constant curvature κ. The second fundamental form is a Codazzi tensor. The Gauss equation S = 2κ − 2a 2 yields finally the result

Deformation of the metric via a Codazzi tensor
In this section we establish some lemmata that we will need later to prove Theorems 1.1 and 1.2. Consider a nondegenerate symmetric (0, 2)-tensor field β on (M n , g) and define a new metric g by (11) g(X, Y ) = g(β(X), β(Y )).
The Levi-Civita connection ∇ of (M n , g) is related to the Levi-Civita connection ∇ of (M n , g) by [6] (12) where Λ is the (1, 2)-tensor field defined by Note that the tensor Λ satisfies g(Λ(X, Z), Y ) + g(Λ(X, Y ), Z) = 0 for all vector fields X, Y, Z. Using formula (12), we can relate the Riemann curvature tensor R of (M n , g) to the one R of (M n , g) by Let (E 1 , . . . , E n ) be a local g-orthonormal frame field on (M n , g). Then the scalar curvature S of (M n , g) is expressed as where Λ ijk := g(Λ(E i , E j ), E k ). We now review briefly the behavior of the Dirac operator under the deformation (11) of metrics. Let Σ(M ) g and Σ(M ) g be the spinor bundles of (M n , g) and (M n , g), respectively. There are natural isomorphisms β −1 : T (M ) −→ T (M ) and β −1 : Σ(M ) g −→ Σ(M ) g preserving the inner products of vectors and spinors as well as the Clifford multiplication: For each spinor field ψ on (M n , g) we denote by ψ := β −1 (ψ) the corresponding spinor field on (M n , g). We will use the same notation for vector fields, X := β −1 (X). It follows from (12) that the spinor derivatives ∇, ∇ are related by Let ω and Ω be a 1-form and a 3-form generated by the tensor Λ via and respectively. The Dirac operator D of (M n , g) can be expressed through the β-twist D β of D as and the square D 2 of the Dirac operator D by In this paper we focus our attention on an interesting property of the tensor Λ. Since we observe that Λ ≡ 0. Consequently, all the equations simplify remarkably when β is a Codazzi tensor. Lemma 2.1. Let β be a nondegenerate Codazzi tensor on (M n , g). Then we have: We close the section with some more lemmata needed in the next section.
Lemma 2.2. Let β be a nondegenerate symmetric tensor field on (M n , g). If there exists a nontrivial spinor field ψ on (M n , g) such that holds for some real-valued functions p, q : M n −→ R and for all vector fields X, then

Proof of the Theorems
Note that the volume form µ of (M n , g) is related to the one µ of (M n , g) by where p, q : M n −→ R are some real-valued functions. Then we have Now, let ϕ = ψ be an eigenspinor of D with eigenvalue λ = 0 ∈ R. By Lemma 2.3, we then see that We choose the functions B, C in such a way that the second integral of (28) vanishes and the equations (23), (24) are satisfied with D β ψ = CDψ. To this end, it is required that the relations hold. Note that (29) implies, in particular, the restriction (4) : so that the last line in the latter part of (28) vanishes. Then we obtain which proves the inequality of Theorem 1.1. The functions optimal for p and q are to be found when considering the limiting case. The former part of (28) yields in the limiting case that Since λ 1 = Cλ 1 , we find that the function C must be a nonzero constant C = c = 0 ∈ R. Then, from (31) and (23), (24), we obtain the two relations in (5) immediately. The condition (6), (7) for the limiting case is easy to check.
To prove Theorem 1.2 we consider the integral and choose the free parameters p, q, F as Then the third line in the latter part of (32) vanishes and we have This proves the inequality (8). The condition (10) for the limiting case is clear.