The Dirac operator on locally reducible Riemannian manifolds

TL;DR

This paper derives sharp estimates for higher eigenvalues of the Dirac operator on locally reducible Riemannian manifolds, relating them to Laplace-Beltrami eigenvalues and scalar curvature, extending known results for the first eigenvalue.

Contribution

It provides new sharp eigenvalue estimates for the Dirac operator on locally reducible manifolds, generalizing previous results for the first eigenvalue.

Findings

Sharp bounds for higher Dirac eigenvalues

Reduction to Alexandrov's result for the first eigenvalue

Relations between Dirac eigenvalues, Laplace-Beltrami eigenvalues, and scalar curvature

Abstract

In this paper, we get estimates on the higher eigenvalues of the Dirac operator on locally reducible Riemannian manifolds, in terms of the eigenvalues of the Laplace-Beltrami operator and the scalar curvature. These estimates are sharp, in the sense that, for the first eigenvalue, they reduce to the result of Alexandrov.

Equations

$$D^2={\nabla }^{\ast }\nabla +\frac{1}{4}ScalId,$$

$${\lambda }^2⩾\frac{1}{4}Sca{l}_{\min },$$

$${\lambda }^2⩾c_{n}Sca{l}_{\min },$$

$${\nabla }_X\psi =-\frac{\lambda }{n}X\cdot \psi ,$$

$${\lambda }_N(D{)}^2\ge \frac{n-1}{n}c{\lambda }_{k+1}(\Delta )+\frac{n-1}{4(n-2)}Sca{l}_{\min },$$

$${\lambda }_N(D{)}^2\ge \frac{n_1-n_2}{n}\frac{n_1}{n_1-1}c{\lambda }_{k+1}(\Delta )+\frac{n_1}{4(n_1-1)}Sca{l}_{\min },$$

$$\begin{array}[]{ccc}\mu:T{M^{n}}\otimes\Sigma{M^{n}}\longrightarrow\Sigma{M^{n}}\\ \ \ \ \ \ \ \ \ \ \ X\otimes\psi\longmapsto X\cdot\psi.\\ \end{array}$$

$$X⟨\varphi ,\phi ⟩$$

$$⟨X\cdot \varphi ,X\cdot \phi ⟩$$

$${\nabla }_X(Y\cdot \varphi )$$

$${\nabla }_{e_k}\psi =e_k(\psi )+\frac{1}{4}{e}_i\cdot {\nabla }_{e_k}{e}_i\cdot \psi$$

$$D\psi$$

$$P\psi$$

$$|\nabla \psi {|}^2=|P\psi {|}^2+\frac{1}{n}|D\psi {|}^2,$$

$${\mathcal{R}}_{X,Y}\psi$$

$$R_{ijkl}\triangleq g(R_{e_i,e_j}{e}_k,e_l)$$

$$e_i\cdot {\mathcal{R}}_{e_j,e_i}\psi =-\frac{1}{2}Ric(e_j)\cdot \psi ,$$

$$D^2={\nabla }^{\ast }\nabla +\frac{1}{4}ScalId,$$

$$g(J(X),J(Y))=g(X,Y),$$

$$J(X)\triangleq X-2g(X,\xi )\xi ,X\in \Gamma (TM)$$

$$D_J\psi \triangleq e_i\cdot {\nabla }_{J(e_i)}\psi =J(e_i)\cdot {\nabla }_{e_i}\psi .$$

$$\eta (X)\triangleq ⟨\varphi ,J(X)\cdot \psi ⟩=g(Y_1,X)+\sqrt{-1}g(Y_2,X),$$

$$div(Y_1)+\sqrt{-1}div(Y_2)=e_i(\eta (e_i))-\eta ({\nabla }_i{e}_i)=-⟨D_J\varphi ,\psi ⟩+⟨\varphi ,D_J\psi ⟩+⟨\varphi ,div(J)\cdot \psi ⟩.$$

$${\lambda }_{\alpha }^2{\int }_M⟨{\psi }_{\alpha },D_J{\psi }_{\beta }⟩$$

$${\lambda }_N({\nabla }^{E\ast }{\nabla }^E+\mathcal{R})\ge (1-c){\lambda }_1(\Delta +{\mathcal{R}}_{min})+c{\lambda }_{k+1}(\Delta +{\mathcal{R}}_{min})$$

$$k=\left[\frac{N}{p+1}\right],c=\frac{1}{8(p+1{)}^2}$$

$${\nabla }_X^{(a,b)}\psi \triangleq {\nabla }_X\psi +aX\cdot \psi +bJ(X)\cdot \psi ,$$

$$X⟨\varphi ,\phi ⟩=⟨{\nabla }_X^{(a,b)}\varphi ,\phi ⟩+⟨\varphi ,{\nabla }_X^{(a,b)}\phi ⟩.$$

$${\nabla }^{(a,b)\ast }{\nabla }^{(a,b)}\psi ={\nabla }^{\ast }\nabla \psi -2aD\psi -2b{D}_J\psi +[2(n-2)ab+n(a^2+b^2)]\psi .$$

$$n_i:=dim\{X\in \Gamma (TM):J(X)=(-1{)}^{i+1}X\}.$$

$${\nabla }_X^{{\lambda }_N}\varphi$$

$${\mathcal{T}}^{{\lambda }_N}$$

Full text

1. Introduction

It is well known that the spectrum of the Dirac operator on closed spin manifolds detects subtle information on the geometry and the topology of such manifolds (see [11]). A fundamental tool to get estimates for eigenvalues of the basic Dirac operator $D$ acting on spinors is the Schrödinger-Lichnerowicz formula

[TABLE]

where $\nabla^{*}$ is the formal adjoint of $\nabla$ with respect to the natural Hermitian inner

product on spinor bundle $\Sigma{M^{n}}$. $Scal$ is the scalar curvature of the closed spin manifold $(M^{n},g)$.

From (2.6), it follows easily that if $\lambda$ is an eigenvalue of $D,$ then

[TABLE]

where $Scal_{\min}\triangleq\min_{M}Scal.$ Clearly, this inequality is interesting only for manifolds with positive scalar curvature, but the minimal value $\frac{1}{4}Scal_{\min}$ cannot be achieved for such manifolds.

The problem of finding optimal lower bounds for the eigenvalues of the Dirac operator on closed manifolds was for the first time considered in 1980 by Friedrich. Using the Lichnerowicz formula and a modified spin connection, he proved the following sharp inequality:

[TABLE]

where $c_{n}=\frac{n}{4(n-1)}.$ The case of equality in (1.2) occurs iff $(M^{n},g)$ admits a nontrivial spinor field $\psi$ called a real Killing spinor, satisfying the following overdetermined elliptic equation

[TABLE]

where $X\in\Gamma(TM)$ and the dot “.” indicates the Clifford multiplication. The manifold must be a locally irreducible Einstein manifold.

The dimension dependent coefficient $c_{n}=\frac{n}{4(n-1)}$ in the estimate can be improved if one imposes geometric assumptions on the metric. Kirchberg [9] showed that for Kähler metrics $c_{n}$ can be replaced by $\frac{n+2}{4n}$ if the complex dimension $\frac{n}{2}$ is odd, and by $\frac{n}{4(n-2)}$ if $\frac{n}{2}$ is even. Alexandrov, Grantcharov, and Ivanov [2] showed that if there exists a parallel one form on $M^{n}$, then $c_{n}$ can be replaced by $c_{n-1}$. Later, Moroianu and Ornea [13] weakened the assumption on the $1$-form from parallel to harmonic with constant length. Note the condition that the norm of the 1-form being constant is essential, in the sense that the topological constraint alone (the existence of a non-trivial harmonic $1$-form) does not allow any improvement of Friedrich’s inequality (see [4]). The generalization of [2] to locally reducible Riemannian manifolds was achieved by Alexandrov [1], extending earlier work by Kim [10].

Another natural way to study the Dirac eigenvalues consists in comparing them with those of other geometric operators. Hijazi’s inequality is already of that kind. As for spectral comparison results between the Dirac operator $D$ and the scalar Laplace operator $\Delta,$ the first ones were proved by Bordoni [5]. They rely on a very nice general comparison principle between two operators satisfying some kind of Kato-type inequality. Bordoni’s results were generalized by Bordoni and Hijazi in the Kähler setting [6]. Recently, with the help of the general spectral result of Bordoni [5] we also obtain that on an $n$-dimensional closed Riemannian spin manifold $(M^{n},g),n\geq 3$ with a non-trivial parallel one form and $Scal\geq 0,$ then for any positive integer $N,$ we have

[TABLE]

where $k=\left[\frac{N}{2^{[\frac{n}{2}]}+1}\right],\ \ c=\frac{1}{8(2^{[\frac{n}{2}]}+1)^{2}}.$

Here, we shall deal with a more general case, and obtain the following estimate on locally reducible Riemannian spin manifolds, which generalizes the result of [1] to arbitray eigenvalue $\lambda_{N}(D).$

Theorem 1**.**

Let $M$ be a locally reducible Riemannian spin compact manifolds with positive scalar. Suppose $TM=T_{1}\oplus\cdots\oplus T_{k},$ where $T_{i}$ are parallel distributions of dimension $n_{i},i=1,\cdots,k,$ and $n_{1}>n_{2}\geq\cdots\geq n_{k}.$ Then for any positive integer N,

[TABLE]

where $k=\left[\frac{N}{2^{[\frac{n}{2}]}+1}\right],c=\frac{1}{8(2^{[\frac{n}{2}]}+1)^{2}}.$

The paper is organised as follows: In Section 2, some preliminaries and lemmas about the Dirac operator $D$ and $J$-twist $D_{J}$ of are given. In Section 3, we obtain the estimates for higher eigenvalues of the Dirac operator on locally decomposable Riemannian spin manifolds. In the end, based on the results in [1] and Section 3, the proof of Theorem 1 are given.

2. the Dirac operator $D$ and the $J$-twist $D_{J}$

We suppose that $(M^{n},g)$ is a closed Riemannian manifold with a fixed spin structure. We understand the spin structure as a reduction $Spin{M^{n}}$ of the SO($n$)-principal bundle of $M^{n}$ to the universal covering $Ad:Spin(n)\rightarrow SO(n)$ of the special orthogonal group. The spinor bundle $\Sigma{M^{n}}=Spin{M^{n}}\times_{\rho}\Sigma_{n}$ on ${M^{n}}$ is the associated complex $2^{[\frac{n}{2}]}$ dimensional complex vector bundle, where $\rho$ is the complex spinor representation. The tangent bundle $T{M^{n}}$ can be regarded as $T{M^{n}}=Spin{M^{n}}\times_{Ad}\mathbb{R}^{n}$. Consequently, the Clifford multiplication on $\Sigma{M^{n}}$ is the fibrewise action given by

[TABLE]

On the spinor bundle $\Sigma{M^{n}}$, one has a natural Hermitian metric, denoted as the Riemannian metric by $\langle\cdot,\cdot\rangle$. The spinorial connection on the spinor bundle induced by the Levi-Civita connection $\nabla$ on ${M^{n}}$ will also be denoted by $\nabla$. The Hermitian metric $\langle\cdot,\cdot\rangle$ and spinorial connection $\nabla$ are compatible with the Clifford multiplication $\mu$. That is

[TABLE]

for $\forall\phi,\varphi\in\Gamma(\Sigma{M^{n}})$ and $\forall X,Y\in\Gamma(T{M^{n}}).$ Using a local orthonormal frame field $\{e_{1},\cdots,e_{n}\}$, the spinorial connection $\nabla$, the Dirac operator $D$ and the twistor operator $P$, are locally expressed as

[TABLE]

and

[TABLE]

which satisfy the following important relation

[TABLE]

for any $\psi\in\Gamma(\Sigma{M^{n}}).$ (Throughout this paper, the Einstein summation notation is always adopted.)

Let $R_{X,Y}Z\triangleq(\nabla_{X}\nabla_{Y}-\nabla_{Y}\nabla_{X}-\nabla_{[X,Y]})Z$ be the Riemannian curvature of $(M^{n},g)$ and denote by $\mathcal{R}_{X,Y}\psi\triangleq(\nabla_{X}\nabla_{Y}-\nabla_{Y}\nabla_{X}-\nabla_{[X,Y]})\psi$ the spin curvature in the spinor bundle $\Sigma{M^{n}}$. They are related via the formula

[TABLE]

We also use the notation

[TABLE]

and $R_{ij}=\langle Ric(e_{i}),e_{j}\rangle\triangleq R_{ikkj},Scal=R_{ii}$. With the help of the Bianchi identity, (2.4) implies

[TABLE]

which in turn gives $2e_{i}\cdot e_{j}\cdot\mathcal{R}_{e_{i},e_{j}}\psi=Scal\ \psi$. Hence one derives the well-known Schrödinger-Lichnerowicz formula

[TABLE]

where $\nabla^{*}$ is the formal adjoint of $\nabla$ with respect to the natural Hermitian inner

product on $\Sigma{M^{n}}$. The formula shows the close relation between $Scal$ and the Dirac operator $D$.

Let $(M^{n},g)$ be an oriented $n$-Riemannian manifold. Let $J$ be a $(1,1)$-tensor field on $(M^{n},g)$ such that $J^{2}=\sigma Id,\sigma=\pm 1$ and

[TABLE]

for all vector fields $X,Y\in\Gamma(T{M^{n}})$ (Here $Id$ stands for the identity map). We say $(M^{n},g,J)$ is an almost Hermitian manifold if $\sigma=-1$ and an almost product Riemannian manifold if $\sigma=1$, respectively. Moreover if $\sigma=-1$ and $J$ is parallel, $(M^{n},g,J)$ is called a Kähler manifold. Similarly, we have the following definition.

Definition 1**.**

[14, 10]** An $n$-Riemannian manifold $(M^{n},g)$ is called locally decomposable if it is an almost product Riemannian manifold $(M^{n},g,J)$ and $J$ is parallel.

Example 1**.**

Suppose an $n$-Riemannian manifold $(M^{n},g)$ possessing a unit vector field $\xi\in\Gamma(T{M^{n}})$, then the reflection $J$ defined by

[TABLE]

is an almost product Riemannian structure. Moreover, it is a locally decomposable Riemannian structure if $\xi$ is a parallel vector field.

As in almost Hermitian spin manifolds, we can also define on almost product Riemannian spin manifolds the following $J$-twist $D_{J}$ of the Dirac operator $D$ by

[TABLE]

It is not difficult to check that $D_{J}$ is a formally self-adjoint elliptic operator with respect to $L^{2}$-product, if ${M^{n}}$ is closed and $div{J}\triangleq(\nabla_{e_{i}}J)(e_{i})=0$. In fact, let we define a complex vector field

[TABLE]

then

[TABLE]

Hence the spectrum of $D_{J}$ is discrete and real.

As in the Kählerian case, Kim obtained the following useful lemma

Lemma 1**.**

$D^{2}=D^{2}_{J}$* holds on any locally decomposable Riemannian spin manifold $(M^{n},g,J)$ ( See Prop. 2.1 in [10], or Lemma 1 in [7]).*

As a simple corollary, one has

Corollary 1**.**

Let $(M^{n},g,J)$ be a locally decomposable Riemannian spin manifold. If $\psi_{\alpha}\in E_{\lambda_{\alpha}}(D)\triangleq\{\psi\neq 0:D\psi=\lambda_{\alpha}\psi\}$ , then $D_{J}\psi_{\alpha}\in E_{\lambda_{\alpha}}(D)\bigoplus E_{-\lambda_{\alpha}}(D).$

Proof. If $D\psi_{\alpha}=\lambda_{\alpha}\psi_{\alpha}$ and $\lambda_{\alpha}^{2}\neq\lambda_{\beta}^{2}$ then Lemma 1 yields

[TABLE]

therefore $\int_{M}\langle D_{J}\psi_{\alpha},\psi_{\beta}\rangle=\int_{M}\langle\psi_{\alpha},D_{J}\psi_{\beta}\rangle=0.$ That is, $D_{J}\psi_{\alpha}\in E_{\lambda_{\alpha}}(D)\bigoplus E_{-\lambda_{\alpha}}(D).$ Q.E.D.

3. locally decomposable Riemannian manifold

We first recall a general spectral comparison result due to Bordoni.(see [5], Theorems 3.2 and 3.3).

Theorem 2**.**

[5]** Let $(M^{n},g)$ be a closed Riemannian manifold. Let E be any vector bundle of rank $p$ on $M,$ endowed with a Hermitian inner product $\langle\cdot,\cdot\rangle$ and a compatible connection $\nabla^{E}$. Let $\mathcal{{R}}$ be any field of symmetric endomorphisms of the fibers, and define the scalar $R_{min}(x)$ as the minimal eigenvalue of $\mathcal{R}_{x}$ acting on $E_{x}$. Then, for any positive integer $N,$ we have:

[TABLE]

where

[TABLE]

and $\Delta$ is the Laplace-Beltrami operator acting on functions.

We shall make use of a modified connection $\nabla^{E}=\nabla^{(a,b)}$ acting on the spinor bundle $\Gamma(\Sigma{M^{n}})$ of a spin manifold admitting a locally decomposable Riemannian structure, to which we apply Theorem 2. For any couple $a,b$ of nontrivial real numbers, define the connection $\nabla^{(a,b)}$ by

[TABLE]

where $J$ is the locally decomposable Riemannian structure. Then, it is easy to check that $\nabla^{(a,b)}$ is compatible with the Hermitian inner product on the spinor bundle. That is, for any vector field $X,$ $\forall\phi,\varphi\in\Gamma(\Sigma{M^{n}})$ one has

[TABLE]

and the rough Laplacian of the modified connection $\nabla^{(a,b)}$ is given by

[TABLE]

Suppose $(M^{n},g,J)$ is a locally decomposable Riemannian spin manifold with $n_{1}>n_{2},$ where

[TABLE]

Let $\lambda_{\alpha}\triangleq\lambda_{\alpha}(D)$ and consider the first $N$ nonnegative eigenvalues, $0\leq\lambda_{1}\leq\lambda_{2}\leq\cdots\leq\lambda_{N}$ and $D\psi_{\alpha}=\lambda_{\alpha}\psi_{\alpha},\alpha=1,\cdots,N.$ Let $a=b=\frac{1}{2n_{1}}\lambda_{N},$ and for $X\in TM^{n}$ and $\phi\in\Gamma(\Sigma M^{n})$

[TABLE]

Hence, if we define the operator

[TABLE]

it follows that

[TABLE]

In particular, for any eigenspinor $\psi_{\alpha}$ such that $D\psi_{\alpha}=\lambda_{\alpha}\psi_{\alpha},\alpha=1,\cdots,N,$ we have

[TABLE]

Next we begin to estimate the $\int_{M}\langle\mathcal{T}^{\lambda_{N}}\psi_{\alpha},\psi_{\alpha}\rangle$ from above.

Proposition 1**.**

Suppose $(M^{n},g,J),n\geq 3$ is a locally decomposable Riemannian spin manifold and $Scal\geq 0,$ then for any positive integer N, we have

[TABLE]

Proof. Motivated by the “twistor-like” operator defined by Moroianu and Ornea in [13], one can define for an any given eigenvalue $\lambda$ the following operator

[TABLE]

Then

[TABLE]

where we choose $J(e_{i})=-e_{i},$ for $i\geq n_{1}+1.$ So for eigenspinor $D\psi_{\alpha}=\lambda_{\alpha}\psi_{\alpha},$ one

[TABLE]

Note (LHS) of the above equality is nonnegative, in fact it can be seen in the following: for any spinorfield $\psi,$

[TABLE]

And one also has for $i\geq n_{1}+1,T_{i}\psi=0$ and

[TABLE]

In particular, this gives for eigenspinor $\psi_{\alpha}$

[TABLE]

That is

[TABLE]

Q.E.D.

Hence with the help of the proposition above, it is not difficult to obtain the following

Theorem 3**.**

Suppose $(M^{n},g,J),n\geq 3$ is a locally decomposable Riemannian spin manifold and $Scal\geq 0,$ then for any positive integer N, we have

[TABLE]

where $k=\left[\frac{N}{2^{[\frac{n}{2}]}+1}\right],\ \ c=\frac{1}{8(2^{[\frac{n}{2}]}+1)^{2}}.$

Proof. Our strategy is to apply Theorem 2 for the modified connection $\nabla^{\lambda_{N}}.$ First, let $E_{N}=L(\psi_{1},\cdots,\psi_{N})$ and $F_{N}$ any $N$-dimensional vector subspace of $\Gamma(\Sigma M).$ Then the min-max principle yields

[TABLE]

where we used Lemma 2 and the key proposition above. On the other hand, Theorem 2 implies that

[TABLE]

so we are done. Q.E.D.

4. on locally reducible Riemannian manifolds

Let $M$ be a compact Riemannian spin manifold with positive scalar $Scal.$ Suppose $TM=T_{1}\oplus\cdots\oplus T_{k},$ where $T_{i}$ are parallel distributions of dimension $n_{i},i=1,\cdots,k,$ and $n_{1}>n_{2}\geq\cdots\geq n_{k}.$ Then one can define a locally decomposable Riemannian structure $J$ as follows

[TABLE]

Moreover, we define the following modified metric connection and corresponding self-adjoint operator for $X\in\Gamma(TM^{n})$ and $\phi\in\Gamma(\Sigma M^{n})$

[TABLE]

Compute for any eigenspinor $\psi_{\alpha}$ such that $D\psi_{\alpha}=\lambda_{\alpha}\psi_{\alpha},\alpha=1,\cdots,N,$

[TABLE]

But [1] gives

[TABLE]

where $\varepsilon_{i}\triangleq\frac{n_{1}}{n_{i}}-1>0,\varepsilon_{2}\leq\cdots\leq\varepsilon_{k},i=2,\cdots,k$ and $D_{i}$ is the “partial” Dirac operator of subbundle $T_{i}.$ Hence

[TABLE]

Suppose

[TABLE]

is an adapted local orthnomal frame, i.e., such that $\{e_{n_{1}+n_{2}+\cdots+n_{i-1}+1},\cdots,e_{n_{1}+n_{2}+\cdots+n_{i}}\}$ spans $T_{i}.$ Let $\alpha_{i}=n_{1}+n_{2}+\cdots+n_{i},$ then

[TABLE]

Note for $i\neq j,$ one has

[TABLE]

that is,

[TABLE]

which, in turn, implies that

[TABLE]

Hence this, together with the inequality (4.5) implies

[TABLE]

This yields that

[TABLE]

Theorem 4**.**

Let $M$ be a compact Riemannian spin manifold with positive scalar $Scal.$ Let $TM=T_{1}\oplus\cdots\oplus T_{k},$ where $T_{i}$ are parallel distributions of dimension $n_{i},i=1,\cdots,k,$ and $n_{1}>n_{2}\geq\cdots\geq n_{k}.$ Then for any positive integer N,

[TABLE]

where $k=\left[\frac{N}{2^{[\frac{n}{2}]}+1}\right],c=\frac{1}{8(2^{[\frac{n}{2}]}+1)^{2}}.$

Proof. As before, the min-max principle yields

[TABLE]

so we are done.