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Limiting Case of the Spin Hypersurface Dirac Operator arising in the positive mass theorem for black holes

Published Online: 31 Dec 2020
Page range: 459 - 466
Received: 04 Jul 2019
Accepted: 30 Sep 2019
Journal Details
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Format
Journal
First Published
01 Jan 2016
Publication timeframe
2 times per year
Languages
English
Abstract

In this paper, we give an explicit form of the scalar curvaure for the limiting case of the eigenvalue of the hypersurface Dirac operator which arises in the positive mass theorem for black holes. Then, we show that the hypersurface is an Einstein.

Keywords

MSC 2010

Introduction

On a compact Riemannian Spin–manifold, mathematicians and physicists have been investigated the spectrum of the Dirac operator to obtain subtle information about the topology and geometry of the manifold and its hypersurface [2,3,4, 6, 10, 13, 14, 17].

In 1963 A. Lichnerowicz proved with the help of the Lichnerowicz–formula that, on a compact Riemannian Spin–manifold (M,g), for any eigenvalue λ of the Dirac operator the inequality λ214infMR {\lambda ^2} \ge {1 \over 4}\mathop {inf}\limits_M R holds [16]. Here R is the scalar curvature of M. The proof of the above inequality is based on the classical spinorial Levi–Civita connection. Then, by using a modified spinorial Levi–Civita connection, T. Friedrich [5] improved the estimate given in (1) for dimension n ≥ 2 as follows λ2n4(n1)infMR. {\lambda ^2} \ge {n \over {4(n - 1)}}\mathop {inf}\limits_M R. Later on, an interesting topological lower bound estimation is given by C. Bär in two dimensional manifold as follows λ22πχ(M)Area(M,g), {\lambda ^2} \ge {{2\pi \chi (M)} \over {Area(M,g)}}, where χ(M) is an Euler–Poincare characteristic of M [1].

After this point, O. Hijazi added a new geometric invariants to obtain an optimal lower bound for the square of the eigenvalue of the Dirac operator. The author improved inequality (2) by using the conformal covariance of the Dirac operator on a compact Riemannian Spin–manifold (M,g) of dimension n ≥ 3, λ2n4(n1)μ1, {\lambda ^2} \ge {n \over {4(n - 1)}}{\mu _1}, where μ1 is the first eigenvalue of the Yamabe operator L given by L:=4n1n2Δg+R L: = 4{{n - 1} \over {n - 2}}{\Delta _g} + R and Δg is the positive Laplacian acting on functions [9]. Also, in 1995 O. Hijazi [11] modified the spinorial Levi–Civita connection in terms of the Energy–Momentum tensor QΨ to obtain the following lower bound λ2infM(R4+|QΨ|2). {\lambda ^2} \ge \mathop {inf}\limits_M \left( {R \over 4} + |{Q_\Psi }{|^2} \right). In addition, in the limiting case of (6) O. Hijazi obtained the following relations (tr(QΨ))2=14R+|QΨ|2,grad(tr(QΨ))=div(QΨ), \matrix{ {{{(tr({Q_\Psi }))}^2} = {1 \over 4}R + |{Q_\Psi }{|^2},} \hfill \cr {grad(tr({Q_\Psi })) = - div({Q_\Psi }),} \hfill \cr } where tr(QΨ) is the trace of Energy–Momentum tensor QΨ.

Using the conformal covariance of the Dirac operator on a compact Riemannian Spin–manifold (M,g), he proved that, any square of the eigenvalue λ of the Dirac operator D satisfies λ2{14μ1+infM|QΨ|2,ifn3,πχ(M)Area(M,g)+infM|QΨ|2,ifn=2, {\lambda ^2} \ge \left\{ \matrix{ {{1 \over 4}{\mu _1} + \mathop {inf}\limits_M |{Q_\Psi }{|^2},} & {\;if\;n \ge 3,} \cr {{{\pi \chi (M)} \over {Area(M,g)}} + \mathop {inf}\limits_M |{Q_\Psi }{|^2},} & {\;if\;n = 2,} \cr} \right. where μ1 is the first eigenvalue of the Yamabe operator L.

After that, on a compact Spin–hypersurface similar studies has begun [12, 18]. Zhang obtain an estimates for the eigenvalue of the operator D˜*D˜ {\widetilde D^*}\widetilde D defined in [18] in terms of the mean curvature and scalar curvature λsupαinfM(Rnα22α+1(n1)(1nα)2|H|2), \lambda \ge \mathop {sup}\limits_\alpha \;\mathop {inf}\limits_M \left({R \over {n{\alpha ^2} - 2\alpha + 1}} - {{(n - 1)} \over {{{(1 - n\alpha )}^2}}}|H{|^2}\right), where α is any real number, α1n \alpha \ne {1 \over n} if H ≠ 0. On a compact Spin–hypersurface MnNn+1 of dimension n ≥ 2, using conformal deformations of the metric, Hijazi and Zhang improved (9) for the eigenvalue of DH (i.e. λH2 \lambda _H^2 is an eigenvalue of D˜*D˜ {\widetilde D^*}\widetilde D ) λsupα,uinfM(R¯e2unα22α+1(n1)(1nα)2|H|2), \lambda \ge \mathop {sup}\limits_{\alpha ,u} \;\mathop {inf}\limits_M \left({{\overline R {e^{2u}}} \over {n{\alpha ^2} - 2\alpha + 1}} - {{(n - 1)} \over {{{(1 - n\alpha )}^2}}}|H{|^2}\right), where is the scalar curvature of M associated to a conformal deformation of metric g and for some real–valued function on N [12]. Moreover, they investigated the limiting case of the above inequality and they show that the hypersurface is an Einstein. Also, they obtain the following estimates for the eigenvalue of the hypersurface Dirac operator Dp defined in [12], λ2{14supb,uinfM(R¯e2u1+nb22b(n1)(1nb)2|P|2);14(nn1μ1supM|P|2)forn3;14(16πArea(M)supM|P|2)n=2,genus=0, {\lambda ^2} \ge \left\{ {\matrix{ {{1 \over 4}\mathop {sup}\limits_{b,u} \;\mathop {inf}\limits_M \left({{\overline R {e^{2u}}} \over {1 + n{b^2} - 2b}} - {{(n - 1)} \over {{{(1 - nb)}^2}}}|P{|^2}\right);} \hfill \cr {{1 \over 4}\left(\sqrt {{n \over {n - 1}}{\mu _1}} - \mathop {sup}\limits_M |P{|^2}\right)\;{\rm{for}}\;n \ge 3;} \hfill \cr {{1 \over 4}\left(\sqrt {{{16\pi } \over {Area(M)}}} - \mathop {sup}\limits_M |P{|^2}\right)\;n = 2,\;{\rm{genus = 0}},} \hfill\cr } } \right. where b is any real–valued function on N.

This paper is organized as follows. At first, we introduce some basic facts concerning hypersurface Dirac operator. Then, we show that the hypersurface manifold is Einstein manifold with constant Ricci curvature by considering the limiting case of the above inequality.

Hypersurface Dirac Operator

Let (N,gN, p) be an (n + 1)–dimensional compact Riemannian Spin–manifold with metric tensor gN, 2–tensor p and M be an n–dimensional Spin–hypersurface in N with its induced metric g. On a compact Riemannian Spin–manifold N, one can construct a spinor bundle denoted by 𝕊 and globally defined along M called hypersurface spinor bundle of M [18]. Let ˜ \widetilde \nabla be the Levi–Civita connection of N, and ∇ be its induced connection on M. Accordingly, ˜ \widetilde \nabla and ∇ can be lifted to the hypersurface spinor bundle 𝕊 and denoted by the same symbol. The Dirac operator D of M defined by ∇ on 𝕊 and the hypersurface Dirac operator D˜ \widetilde D by ˜ \widetilde \nabla on 𝕊.

Recall that, 𝕊 carries a natural positive definite Hermitian metric on Γ(𝕊) denoted by ( , ) and satisfies, for any covector field vT*N, and any spinor fields Ψ, Φ ∈ Γ(𝕊) (vΨ,vΦ)=|v|2(Ψ,Φ). (v \cdot \Psi ,v \cdot \Phi ) = |v{|^2}(\Psi ,\Phi ). This metric is globally–defined along M. Also, ˜ \widetilde \nabla and its induced connection ∇ are compatible with the Hermitian metric ( , ) [15].

Let xM be a fixed point and eα be an orthonormal basis of TxN with e0 normal to M and ei tangent to M such that for 1 ≤ i, jn, (iej)x=(˜0ej)x=0, {({\nabla _i}{e_j})_x} = ({\widetilde \nabla _0}{e_j}{)_x} = 0, Moreover, let eα be the dual coframe of eα. Then, for 1 ≤ i, jn (˜iej)x=hije0,(˜ie0)x=hijej, \matrix{ {{{({{\widetilde \nabla }_i}{e^j})}_x} = - {h_{ij}}{e^0},} \hfill \cr {{{({{\widetilde \nabla }_i}{e^0})}_x} = {h_{ij}}{e^j},} \hfill \cr } where hij=(˜ie0,ej) {h_{ij}} = ({\widetilde \nabla _i}{e_0},{e_j}) are the components of the second fundamental form at x. Let P be a function defined as [12] P:=gNijpij|M. P: = g_N^{ij}{p_{ij}}{|_M}. Then, the hypersurface Dirac operator which is arises in the positive mass theorem for black holes defined as follows [7, 8, 13, 19, 20]. DHP=e0DH212Pe0. {D_{HP}} = {e^0} \cdot D - {H \over 2} - {{\sqrt { - 1} } \over 2}P{e^0} \cdot .

As in [12], in this paper we consider the hypersurface dirac operator Dp defined as DP:=e0D12Pe0. {D_P}: = {e^0} \cdot D - {{\sqrt { - 1} } \over 2}P{e^0} \cdot . In the following, we consider limiting case of λp.

Limiting Case of the Hypersurface manifold endowed with Spin Structure

In this section, by using the modified spinorial Levi–Civita connection used to obtain the lower bound of the eigenvalue of the hypersurface Dirac operator arises in the positive mass theorem for black holes, we obtain the scalar curvature corresponding to the limiting case of the eigenvalue of the hypersurface Dirac operator. Then, we show that the hypersurface manifold (M,g) is an Einstein manifold. Finally, we give an explicit form of the eigenvalue in the limiting case.

In the following theorem, to obtain this lower bound we consider the following modified spinorial Levi–Civita connection defined as [12]: ibΨ=iΨ+(1b2(1nb))1PeiΨbeie0DpΨ. \nabla _i^b\Psi = {\nabla _i}\Psi + \left({{1 - b} \over {2(1 - nb)}}\right)\sqrt { - 1} P{e^i} \cdot \Psi - b{e^i} \cdot {e^0} \cdot {D_p}\Psi .

Theorem 1

Let MN be a compact Riemannian Spin–hypersurface. If λp achieves its minimum, M is an Einstein manifold and it has constant Ricci curvature and constant P. Also the following holds: R=n(n1)(nb022b0+1)2(1nb0)4P,λp2=14supb0(n1)2(1nb0)4P,Rik=(n1)(nb022b0+1)2(1nb0)4Pδik, \matrix{ \,\,\;{R = n(n - 1){{{{(nb_0^2 - 2{b_0} + 1)}^2}} \over {{{(1 - n{b_0})}^4}}}P,} \hfill \cr \,\;{\lambda _p^2 = {1 \over 4}\mathop {sup}\limits_{{b_0}} {{{{(n - 1)}^2}} \over {{{(1 - n{b_0})}^4}}}P,} \hfill \cr {{R_{ik}} = (n - 1){{{{(nb_0^2 - 2{b_0} + 1)}^2}} \over {{{(1 - n{b_0})}^4}}}P{\delta _{ik}},} \hfill \cr } where b0 is chosen such that the right side of (11) achieves its maximum.

Proof

If λp achieves its minimum, then ibΨ0 \nabla _i^b\Psi \equiv 0 . This implies iΨ=(1b02(1nb0))1PeiΨ+b0eie0DpΨ, {\nabla _i}\Psi = - \left({{1 - {b_0}} \over {2(1 - n{b_0})}}\right)\sqrt { - 1} P{e^i} \cdot \Psi + {b_0}{e^i} \cdot {e^0} \cdot {D_p}\Psi , where b is any real–valued function on M. Performing its Clifford multiplication by ei, for any spinor field Ψ ∈ Γ(𝕊) and for any i,1 ≤ in, yields eiiΨ=(1b02(1nb0))1PΨb0e0DpΨ. {e^i} \cdot {\nabla _i}\Psi = \left({{1 - {b_0}} \over {2(1 - n{b_0})}}\right)\sqrt { - 1} P\Psi - {b_0}{e^0} \cdot {D_p}\Psi . Summing over i, gives DΨ=n(1b02(1nb0))1PΨnb0e0DpΨ=n(1b02(1nb0))1PΨn0be0(e0DΨ12Pe0Ψ)=n(1b02(1nb0))1PΨ+nb0DΨnb012PΨ=(n2b022b0n+n2(1nb0))1PΨ+nb0DΨ. \matrix{ {D\Psi = n\left({{1 - {b_0}} \over {2(1 - n{b_0})}}\right)\sqrt { - 1} P\Psi - n{b_0}{e^0} \cdot {D_p}\Psi } \hfill \cr {\;\;\;\;\;\; = n\left({{1 - {b_0}} \over {2(1 - n{b_0})}}\right)\sqrt { - 1} P\Psi - {n_0}b{e^0} \cdot ({e^0} \cdot D\Psi - {{\sqrt { - 1} } \over 2}P{e^0} \cdot \Psi )} \hfill \cr {\;\;\;\;\;\; = n\left({{1 - {b_0}} \over {2(1 - n{b_0})}}\right)\sqrt { - 1} P\Psi + n{b_0}D\Psi - n{b_0}{{\sqrt { - 1} } \over 2}P\Psi } \hfill \cr {\;\;\;\;\;\; = \left({{{n^2}b_0^2 - 2{b_0}n + n} \over {2(1 - n{b_0})}}\right)\sqrt { - 1} P\Psi + n{b_0}D\Psi .} \hfill \cr } As a result, we get DΨ=(n2b022b0n+n2(1nb0)2)1PΨ. D\Psi = ({{{n^2}b_0^2 - 2{b_0}n + n} \over {2(1 - n{b_0}{)^2}}})\sqrt { - 1} P\Psi . Again, considering (19), we obtain iΨ=(1b02(1nb0))1PeiΨ+b0eie0(e0DΨ12Pe0Ψ)=(1b02(1nb0))1PeiΨ+b0ei(DΨ+12PΨ)=(1b02(1nb0))1PeiΨ+b0ei((n2b022b0n+n2(1nb0)2)1PΨ+12PΨ)=(nb022b0+12(1nb0)2)1PeiΨ=1P˜eiΨ, \matrix{ {{\nabla _i}\Psi = - \left({{1 - {b_0}} \over {2(1 - n{b_0})}}\right)\sqrt { - 1} P{e^i} \cdot \Psi + {b_0}{e^i} \cdot {e^0} \cdot \left({e^0} \cdot D\Psi - {{\sqrt { - 1} } \over 2}P{e^0} \cdot \Psi \right)} \hfill \cr {\;\;\;\;\;\;\; = - \left({{1 - {b_0}} \over {2(1 - n{b_0})}}\right)\sqrt { - 1} P{e^i} \cdot \Psi + {b_0}{e^i} \cdot \left( - D\Psi + {{\sqrt { - 1} } \over 2}P\Psi \right)} \hfill \cr {\;\;\;\;\;\;\; = - \left({{1 - {b_0}} \over {2(1 - n{b_0})}}\right)\sqrt { - 1} P{e^i} \cdot \Psi + {b_0}{e^i} \cdot \left( - ({{{n^2}b_0^2 - 2{b_0}n + n} \over {2(1 - n{b_0}{)^2}}}\right)\sqrt { - 1} P\Psi + {{\sqrt { - 1} } \over 2}P\Psi )} \hfill \cr {\;\;\;\;\;\;\; = - \left({{nb_0^2 - 2{b_0} + 1} \over {2(1 - n{b_0}{)^2}}}\right)\sqrt { - 1} P{e^i} \cdot \Psi } \hfill \cr {\;\;\;\;\;\;\; = \sqrt { - 1} \widetilde P{e^i} \cdot \Psi ,} \hfill \cr } where P˜=(nb022b0+12(1nb0)2)P \widetilde P = - ({{nb_0^2 - 2{b_0} + 1} \over {2(1 - n{b_0}{)^2}}})P . Accordingly, k,l14RijklekelΨ=(jiij)Ψ=j(1P˜eiΨ)i(1P˜ejΨ)=1jP˜eiΨ+1P˜eijΨ1iP˜ejΨ1P˜ejiΨ. \matrix{ {\sum\limits_{k,l} {1 \over 4}{R_{ijkl}}{e^k} \cdot {e^l} \cdot \Psi = ({\nabla _j}{\nabla _i} - {\nabla _i}{\nabla _j})\Psi } \hfill \cr {\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\; = {\nabla _j}(\sqrt { - 1} \widetilde P{e^i} \cdot \Psi ) - {\nabla _i}(\sqrt { - 1} \widetilde P{e^j} \cdot \Psi )} \hfill \cr {\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\; = \sqrt { - 1} {\nabla _j}\widetilde P{e^i} \cdot \Psi + \sqrt { - 1} \widetilde P{e^i} \cdot {\nabla _j}\Psi - \sqrt { - 1} {\nabla _i}\widetilde P{e^j} \cdot \Psi - \sqrt { - 1} \widetilde P{e^j} \cdot {\nabla _i}\Psi .} \hfill \cr } Clifford multiplication with ej, we obtain k12RikekΨ=j,k,l14RijklejekelΨ=1jP˜ejeiΨ+1P˜ejeijΨ1iP˜ejejΨ1P˜ejejiΨ=1jP˜ejeiΨ+1P˜ejeijΨ+n1iP˜Ψ+n1P˜iΨ=1jP˜(eiej2δij)Ψ+1P˜(eiej2δij)jΨ+n1iP˜Ψ+n1P˜iΨ. \matrix{ {\sum\limits_k {1 \over 2}{R_{ik}}{e^k} \cdot \Psi = \sum\limits_{j,k,l} {1 \over 4}{R_{ijkl}}{e^j} \cdot {e^k} \cdot {e^l} \cdot \Psi } \hfill \cr \;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;{ = \sqrt { - 1} {\nabla _j}\widetilde P{e^j} \cdot {e^i} \cdot \Psi + \sqrt { - 1} \widetilde P{e^j} \cdot {e^i} \cdot {\nabla _j}\Psi - \sqrt { - 1} {\nabla _i}\widetilde P{e^j} \cdot {e^j} \cdot \Psi } { - \sqrt { - 1} \widetilde P{e^j} \cdot {e^j} \cdot {\nabla _i}\Psi } \hfill \cr \;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;{ = \sqrt { - 1} {\nabla _j}\widetilde P{e^j} \cdot {e^i} \cdot \Psi + \sqrt { - 1} \widetilde P{e^j} \cdot {e^i} \cdot {\nabla _j}\Psi + n\sqrt { - 1} {\nabla _i}\widetilde P\Psi } { + n\sqrt { - 1} \widetilde P{\nabla _i}\Psi } \hfill \cr \;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\; { = \sqrt { - 1} {\nabla _j}\widetilde P( - {e^i} \cdot {e^j} \cdot - 2{\delta _{ij}})\Psi + \sqrt { - 1} \widetilde P( - {e^i} \cdot {e^j} \cdot - 2{\delta _{ij}}){\nabla _j}\Psi } { + n\sqrt { - 1} {\nabla _i}\widetilde P\Psi + n\sqrt { - 1} \widetilde P{\nabla _i}\Psi .} \hfill \cr } Again, Clifford multiplication with ei, we have 12RΨ=i,k12RikeiekΨ=1jP˜ei(eiej2δij)Ψ+1P˜ei(eiej2δij)jΨ+n1eiiP˜Ψ+n1P˜eiiΨ=n1jP˜ejΨ21jP˜ejΨ+n1P˜ejjΨ21P˜ejjΨ+n1eiiP˜Ψ+n1P˜eiiΨ=2(n1)1dP˜Ψ+2(n1)1P˜DΨ=2(n1)1dP˜Ψ2n(n1)1P˜(1P˜Ψ)=2(n1)1dP˜Ψ2n(n1)P˜2Ψ. \matrix{ { - {1 \over 2}R\Psi = \sum\limits_{i,k} {1 \over 2}{R_{ik}}{e^i} \cdot {e^k} \cdot \Psi } \hfill \cr {\;\;\;\;\;\;\;\;\;\;\; = \sqrt { - 1} {\nabla _j}\widetilde P{e^i} \cdot ( - {e^i} \cdot {e^j} \cdot - 2{\delta _{ij}})\Psi + \sqrt { - 1} \widetilde P{e^i} \cdot ( - {e^i} \cdot {e^j} \cdot - 2{\delta _{ij}}){\nabla _j}\Psi + n\sqrt { - 1} {e^i}{\nabla _i}\widetilde P\Psi + n\sqrt { - 1} \widetilde P{e^i} \cdot {\nabla _i}\Psi } \hfill \cr {\;\;\;\;\;\;\;\;\;\;\; = n\sqrt { - 1} {\nabla _j}\widetilde P{e^j} \cdot \Psi - 2\sqrt { - 1} {\nabla _j}\widetilde P{e^j} \cdot \Psi + n\sqrt { - 1} \widetilde P{e^j} \cdot {\nabla _j}\Psi - 2\sqrt { - 1} \widetilde P{e^j} \cdot {\nabla _j}\Psi + n\sqrt { - 1} {e^i}{\nabla _i}\widetilde P\Psi + n\sqrt { - 1} \widetilde P{e^i} \cdot {\nabla _i}\Psi } \hfill \cr {\;\;\;\;\;\;\;\;\;\;\; = 2(n - 1)\sqrt { - 1} d\widetilde P \cdot \Psi + 2(n - 1)\sqrt { - 1} \widetilde PD\Psi } \hfill \cr {\;\;\;\;\;\;\;\;\;\;\; = 2(n - 1)\sqrt { - 1} d\widetilde P \cdot \Psi - 2n(n - 1)\sqrt { - 1} \widetilde P(\sqrt { - 1} \widetilde P\Psi )} \hfill \cr {\;\;\;\;\;\;\;\;\;\;\; = 2(n - 1)\sqrt { - 1} d\widetilde P \cdot \Psi - 2n(n - 1){{\widetilde P}^2}\Psi .} \hfill \cr } Accordingly, P˜ \widetilde P must be constant. Therefore R=4n(n1)P˜2=n(n1)(nb022b0+1)2(1nb0)4P. R = 4n(n - 1){\widetilde P^2} = n(n - 1){{{{(nb_0^2 - 2{b_0} + 1)}^2}} \over {{{(1 - n{b_0})}^4}}}P. In this case, we have λp2=14supb0(n1)2(1nb0)4P. \lambda _p^2 = {1 \over 4}\mathop {sup}\limits_{{b_0}} {{{{(n - 1)}^2}} \over {{{(1 - n{b_0})}^4}}}P. On the other hand, k12RikekΨ=1P˜(eiej2δij)jΨ+n1P˜iΨ=1P˜eiDΨ21P˜iΨ+n1P˜iΨ=nP˜2eiΨ+(n2)P˜2eiΨ=2(n1)P˜2eiΨ. \matrix{ {\sum\limits_k {1 \over 2}{R_{ik}}{e^k} \cdot \Psi = \sqrt { - 1} \widetilde P( - {e^i} \cdot {e^j} \cdot - 2{\delta _{ij}}){\nabla _j}\Psi + n\sqrt { - 1} \widetilde P{\nabla _i}\Psi } \hfill \cr {\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\; = - \sqrt { - 1} \widetilde P{e^i} \cdot D\Psi - 2\sqrt { - 1} \widetilde P{\nabla _i}\Psi + n\sqrt { - 1} \widetilde P{\nabla _i}\Psi } \hfill \cr {\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\; = n{{\widetilde P}^2}{e^i} \cdot \Psi + (n - 2){{\widetilde P}^2}{e^i} \cdot \Psi } \hfill \cr {\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\; = 2(n - 1){{\widetilde P}^2}{e^i} \cdot \Psi .} \hfill \cr } Therefore Rik=4(n1)P˜2δik=(n1)(nb022b0+1)2(1nb0)4Pδik. \matrix{ {{R_{ik}} = 4(n - 1){{\widetilde P}^2}{\delta _{ik}}} \hfill \cr {\;\;\;\;\; = (n - 1){{{{(nb_0^2 - 2{b_0} + 1)}^2}} \over {{{(1 - n{b_0})}^4}}}P{\delta _{ik}}.} \hfill \cr }

In this part of the paper, concerning conformal change of the Riemannian metric and using the classic arguments given in [11,12,13], the limiting case of the square of the eigenvalue λp of the hypersurface dirac operator Dp is handled.

Considering the conformal metric g¯N=e2ugN {\overline g _N} = {e^{2u}}{g_N} given with any real–valued function u on N. Accordingly, the Gu isometry defined between SOgN N and SOg¯NN S{O_{{{\overline g }_N}}}N reduces an isometry reduction between the SpingN N and Sping¯NN {Spin_{{{\overline g }_N}}}N principal bundles as well as an isometry between the corresponding hypersurface spinor bundles 𝕊 and S¯(GuS) \overline{\mathbb{S}}(\equiv G_u\mathbb{S}) , respectively [12]. Denote by Ψ¯=GuΨ \overline \Psi = {G_u}\Psi the corresponding sections of S¯ \mathbb{S} , for any spinorfield Ψ of 𝕊. Note that, g¯:=g¯N|M \overline g : = {\overline g _N}{|_M} is induced metric of g¯N {\overline g _N} on M. Also, by using the Clifford multiplication on S¯ \mathbb{S} is given by ei¯¯Ψ¯=eiΨ¯, \overline {{e^i}} \overline \cdot \overline \Psi = \overline {{e^i} \cdot \Psi } , we obtain the following argument [12] p¯i¯j¯=eupij. {\overline p _{\overline i \;\overline j }} = {e^{ - u}}{p_{ij}}. Using p¯i¯j¯=eupij {\overline p _{\overline i \;\overline j }} = {e^{ - u}}{p_{ij}} , we get P¯=g¯Ni¯j¯p¯i¯j¯|M=e2ugN(ei¯,ej¯)p¯i¯j¯|M=gN(ei,ej)eupij|M=euP. \matrix{ {\overline P = \,\overline g _N^{\overline i \;\overline j }\;{{\overline p }_{\overline i \;\overline j }}{|_M}} \hfill \cr {\;\;\; = {e^{2u}}{g_N}(\overline {{e^i}} ,\;\overline {{e^j}} ){{\overline p }_{\overline i \;\overline j }}{|_M}} \hfill \cr {\;\;\; = {g_N}({e^i},{e^j}){e^{ - u}}{p_{ij}}{|_M}} \hfill \cr {\;\;\; = {e^{ - u}}P.} \hfill \cr } Under the conformal change of the Riemannian metric, the modified spinorial Levi–Civita connection (17) is transformed as ¯ei¯bΨ¯=¯ei¯Ψ¯+(1b2(1nb))1P¯ei¯¯Ψ¯bei¯¯e0¯¯D¯pΨ¯. \overline \nabla _{\overline {{e^i}} }^b\overline \Psi = {\overline \nabla _{\overline {{e^i}} }}\overline \Psi + ({{1 - b} \over {2(1 - nb)}})\sqrt { - 1} \;\overline P \;\overline {{e^i}} \;\overline \cdot \;\overline \Psi - b\overline {{e^i}} \;\overline \cdot \;\overline {{e^0}} \;\overline \cdot \;{\overline D _p}\;\overline \Psi .

If λp achieves its minimum, then ¯ei¯bΨ¯=0 \overline \nabla _{\overline {{e^i}} }^b\overline \Psi = 0 . This means that ¯ei¯Ψ¯=(1b02(1nb0))1P¯ei¯¯Ψ¯+b0ei¯¯e0¯¯D¯pΨ¯. {\overline \nabla _{\overline {{e^i}} }}\overline \Psi = - \left({{1 - {b_0}} \over {2(1 - n{b_0})}}\right)\sqrt { - 1} \;\overline P \;\overline {{e^i}} \;\overline \cdot \;\overline \Psi + {b_0}\overline {{e^i}} \;\overline \cdot \;\overline {{e^0}} \;\overline \cdot \;{\overline D _p}\;\overline \Psi . Using the method given in the proof of Theorem (1), we get D¯Ψ¯=(n2b022b0n+n2(1nb0)2)1P¯Ψ¯. \overline D \;\overline \Psi = ({{{n^2}b_0^2 - 2{b_0}n + n} \over {2(1 - n{b_0}{)^2}}})\sqrt { - 1} \;\overline P \;\overline \Psi . Also, (23) transform into ¯ei¯Ψ¯=1P¯˜ei¯¯Ψ¯, {\overline \nabla _{\overline {{e^i}} }}\overline \Psi = \sqrt { - 1} \;\widetilde {\overline P }\;\overline {{e^i}} \;\overline \cdot \;\overline \Psi , where P¯˜=(nb022b0+12(1nb0)2)P¯ \widetilde {\overline P } = - \left({{nb_0^2 - 2{b_0} + 1} \over {2(1 - n{b_0}{)^2}}}\right)\overline P . In addition, scalar curvature of (M,g̅) is obtained as 12R¯Ψ¯=i,k12R¯i¯,k¯ei¯¯ek¯¯Ψ¯=2(n1)1dP¯˜Ψ¯2n(n1)(P¯˜)2Ψ¯. \matrix{ { - {1 \over 2}\overline R \;\overline \Psi = \sum\limits_{i,k} {1 \over 2}{{\overline R }_{\overline i ,\;\overline k }}\overline {{e^i}} \;\overline \cdot \overline {{e^k}} \;\overline \cdot \;\overline \Psi } \hfill \cr {\;\;\;\;\;\;\;\;\;\;\;\; = 2(n - 1)\sqrt { - 1} d\widetilde {\overline P } \cdot \overline \Psi - 2n(n - 1)(\widetilde {\overline P }{)^2}\overline \Psi .} \hfill \cr } Accordingly, P¯˜ \widetilde {\overline P } must be constant. Therefore R¯=4n(n1)P¯˜2=n(n1)(nb022b0+1)2(1nb)4P¯2=e2un(n1)(nb022b0+1)2(1nb0)4P2. \matrix{ {\overline R = 4n(n - 1){{\widetilde {\overline P }}^2} = n(n - 1){{{{(nb_0^2 - 2{b_0} + 1)}^2}} \over {{{(1 - nb)}^4}}}{{\overline P }^2}} \hfill \cr {\,\;\; = {e^{ - 2u}}n(n - 1){{{{(nb_0^2 - 2{b_0} + 1)}^2}} \over {{{(1 - n{b_0})}^4}}}{P^2}.} \hfill \cr } In this case, we have λp2=14supb0(n1)2(1nb0)4P. \lambda _p^2 = {1 \over 4}\mathop {sup}\limits_{{b_0}} {{{{(n - 1)}^2}} \over {{{(1 - n{b_0})}^4}}}P. On the other hand, k12R¯i¯k¯ek¯¯Ψ¯=2(n1)P¯˜2ei¯¯Ψ¯. \sum\limits_k {1 \over 2}{\overline R _{\overline i \;\overline k }}\overline {{e^k}} \overline \cdot \;\overline \Psi = 2(n - 1){\widetilde {\overline P }^2}\overline {{e^i}} \;\overline \cdot \;\overline \Psi . Therefore R¯i¯k¯=4(n1)P¯˜2δi¯k¯=(n1)(nb022b0+1)2(1nb0)4P¯δi¯k¯=(n1)(nb022b0+1)2(1nb0)4e2uP2δi¯k¯. \matrix{ {{{\overline R }_{\overline i \;\overline k }} = 4(n - 1){{\widetilde {\overline P }}^2}{\delta _{\overline i \;\overline k }}} \hfill \cr {\;\;\;\;\;\; = (n - 1){{{{(nb_0^2 - 2{b_0} + 1)}^2}} \over {{{(1 - n{b_0})}^4}}}\overline P {\delta _{\overline i \;\overline k }}} \hfill \cr {\;\;\;\;\;\; = (n - 1){{{{(nb_0^2 - 2{b_0} + 1)}^2}} \over {{{(1 - n{b_0})}^4}}}{e^{ - 2u}}{P^2}{\delta _{\overline i \;\overline k }}.} \hfill \cr } According to above equality, (M,g̅) is an Einstein manifold.

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