DIFFERENTIAL GEOMETRY AND APPLICATIONS Satellite Conference of ICM in Berlin, Aug. 10 – 14, 1998, Brno Masaryk University in Brno (Czech Republic) 1999, 47–60

CURVATURE BOUNDS FOR THE SPECTRUM OF A COMPACT EINSTEIN-LIKE MANIFOLD

SHARIEF DESHMUKH

Abstract. Let (M, g) be a connected compact n-dimensional, Riemannian manifold which is of class A according to Gray [3, p.261]. If the sectional curvatures of M are bounded below by a constant k_{0} and the square of the length of curvature tensor field ∥R_{X}∥^{2} in the direction of a vector field X satisfies ∥R_{X}∥^{2} ≤ 2(n − 1)k_{0}^{2}L∥X∥^{2}, where L is a constant (for the unit nsphere S^{n} it satisfies ∥R_{X}∥^{2} = 2(n − 1)∥X∥^{2}), then it is shown that either M is isometric to a sphere or else each nonzero eigenvalue λ of the Laplacian satisfies one of the following (i) λ > \frac{2(3n−2)}{3} k_{0}, (ii) λ < nk_{0}L, (iii) \frac{2(3n−2)}{3} k_{0} < λ < nk_{0}L.

In addition to this, an estimate on ∥R_{X}∥^{2} is used to characterize spheres among the Riemannian manifolds in the class A.

1. Introduction

One of the interesting questions in the Geometry of a Riemannian manifold (M, g) is to obtain lower bounds for the eigenvalues of the Laplacian operator Δ acting on the smooth functions on M. One of the classical results in this direction states that if the Ricci curvature of (M, g) satisfies Ric ≥ (n − 1)k, k > 0, n = dim M and M is compact, then the first nonzero eigenvalue λ_{1} of Δ satisfies λ_{1} ≥ nk (cf. [1]). Simon [5] considered compact Einstein manifolds (M, g), dim M ≥ 2 of constant scalar curvature (which is automatic for dim M ≥ 3), and proved that either they are isometric to a sphere or that each eigenvalue λ of the Laplacian satisfies λ > 2nk_{0}, where k_{0} is the lower bound of the sectional curvatures. A similar result is obtained by Tanno [6]. Motivated by the work of Simon [5], we consider in this paper a compact Riemannian manifold (M, g) whose Ricci tensor satisfies (∇_{X}Ric)(X, X) = 0, X ∈ χ(M), where χ(M) is the Lie algebra of the smooth vector fields on M and ∇ is the Riemannian connection, and obtain the lower bounds for the eigenvalues of the Laplacian under some curvature

1991 Mathematics Subject Classification. Primary 53C20, 58G25; Secondary 53B25, 53C20, 53C40.

Key words and phrases. Einstein-like manifolds, eigenvalues of Laplacian, isometric to a sphere.

This paper is in final form and no version of it will be submitted for publication elsewhere.

assumptions. These Riemannian manifolds have been studied by Gray [3], where the class of such manifolds is denoted by A and that of Einstein manifolds by E and proved the strict inclusion E ⊂ A (cf. §4 in [3]). In our work we use the square of the length ∥R_{X}∥^{2} of the curvature tensor field in the direction of X defined by

∥R_{X}∥^{2} = ∑_{ij} ∥R(e_{i}, e_{j})X)∥^{2}, X ∈ χ(M)

for a local orthonormal frame {e_{1}, …, e_{n}}. For a sphere S^{n} of constant curvature k_{0} this turns out to be ∥R_{X}∥^{2} = 2k_{0}^{2}(n − 1)∥X∥^{2}. Our main result is the following Theorem. Let (M, g) be an n-dimensional compact and connected Riemannian manifold whose Ricci curvature satisfies

(∇_{X}Ric)(X, X) = 0, X ∈ χ(M) .

If all the sectional curvatures of M are bounded below by a constant k_{0} and the square of the length ∥R_{X}∥^{2} of the curvature tensor field in the direction of X ∈ χ(M) satisfies ∥R_{X}∥^{2} ≤ 2k_{0}^{2}(n−1)∥X∥^{2}L, where L is a constant, then either M is isometric to a sphere or else each eigenvalue λ of the Laplacian satisfies one of the following

(i) λ > \frac{2(3n − 2)}{3}k_{0}, (ii) λ < nk_{0}L, (iii) \frac{2(3n − 2)}{3}k_{0} < λ < nk_{0}L .

We have also shown that an appropriate upper bound of ∥R_{X}∥^{2} in terms of the first nonzero eigenvalue of the Laplacian gives a characterization of spheres among the Riemannian manifolds in the class A (see theorems 4.1 and 4.2).

2. Preliminaries

Throughout this section (M, g) will be an n-dimensional Riemannian manifold without boundary and with Riemannian connection ∇. For a smooth function f : M → R, the Laplacian of f is defined by Δf = −div(grad f).

For a smooth function f : M → R, we define an operator A : χ(M) → χ(M) by A(X) = ∇_{X}grad f, X ∈ χ(M). This operator has the following properties:

tr. A = −Δf, g(AX, Y) = g(X, AY), (2.1) (∇A)(X, Y) = (∇A)(Y, X) + R(X, Y) grad f , (2.2)

where (∇A)(X, Y) = ∇_{X}A(Y)−A(∇_{X}Y) and R is the curvature tensor field given by

R(X, Y)Z = ∇_{X}∇_{Y} Z − ∇_{Y} ∇_{X}Z − ∇_{[X,Y]}Z .

g((∇A)(X, Y), Z) = g(Y, (∇A)(X, Z)), X, Y, Z ∈ χ(M). (2.3)

This follows from the fact that A is symmetric (see (2.1)).

We denote by Q the Ricci operator defined by

Ric(X, Y) = g(Q(X), Y), X, Y ∈ χ(M) . (2.4)

CURVATURE BOUNDS FOR THE SPECTRUM … 49

The second covariant derivative (∇^{2}A)(X, Y, Z), is defined by

(∇^{2}A)(X, Y, Z) = ∇_{X}(∇A)(Y, Z) − (∇A)(∇_{X}Y, Z) − (∇A)(Y, ∇_{X}Z) , (2.5)

and we have the following Ricci identity

\begin{matrix}(∇^{2}A)(X, Y, Z) − (∇^{2}A)(Y, X, Z) = R(X, Y)AZ − A(R(X, Y)Z), \\ X, Y, Z ∈ χ(M) . \end{matrix} (2.6)

Lemma 1.

(∇^{2}A)(X, X, Z) =

(∇^{2}A)(Z, X, X) + 2R(X, Z)AX − A(R(X, Z)X) + (∇_{X}R)(X, Z) grad f ,

where A(X) = ∇_{X} grad f, X ∈ χ(M). Proof. This follows immediately from (2.2), (2.5), (2.6) and the definition of the operator A. Lemma 2. For a local orthonormal frame {e_{1}, …, e_{n}} of M

∑_{i}(∇A)(e_{i}, e_{i}) = −gradΔf + Q(grad f) ,

where A(X) = ∇_{X} grad f, X ∈ χ(M). Proof. Using (2.1), we have −Δf = ∑_{i} g(Ae_{i}, e_{i}), and thus for X ∈ χ(M), we have

−X(Δf) = ∑_{i} g((∇A)(X, e_{i}), e_{i}) = ∑_{i} g((∇A)(e_{i}, X) + R(X, e_{i})grad f, e_{i})
= ∑_{i} g(X, (∇A)(e_{i}, e_{i})) + ∑_{i} g(R(X, e_{i})grad f, e_{i})
= g(∑_{i}(∇A)(e_{i}, e_{i}), X) − Ric(X, grad f),

where we have used (2.2) and (2.3). The last equation proves the Lemma. Lemma 3. Let (M, g) be a compact Riemannian manifold and f : M → R be a smooth function. Then

∫_{M}{(Δf)^{2} − g(grad f, grad△f)}dv = 0 .

Proof. Using Δf = −div(grad f), we obtain

Δ(fΔf) = fΔ^{2}f + (Δf)^{2} − 2g(grad f, gradΔf) .

Integrating this equation and using the fact that Δ is self-adjoint together with Stokes theorem we get the desired result. Lemma 4. Let (M, g) be a compact Riemannian manifold and f : M → R be a smooth function. Then

∫_{M}{∥A∥^{2} − (Δf)^{2} + Ric(grad f, grad f)}dv = 0

where A(X) = ∇_{X} grad f, X ∈ χ(M).

Proof. We use definition of A and equation (2.3) to compute

div(A(grad f)) = ∑_{i} g(∇_{e_{i}}A(grad f), e_{i})
= ∑_{i} g((∇A)(e_{i}, grad f) + A(∇e_{i}grad f), e_{i})
= ∑_{i} g(grad f, (∇A)(e_{i}, e_{i})) + ∑_{i} g(Ae_{i}, Ae_{i})
= g(grad f, ∑_{i}(∇A)(e_{i}, e_{i})) + ∥A∥^{2} .

Integrating this equation and using Lemma 2, we get the result.

Remark. If λ is an eigenvalue of Δ, i.e., Δf = λf, then Lemma 3 gives

∫_{M}{(Δf)^{2} − λ∥grad f∥}dv = 0. Using this equation and the Schwarz inequality

∥A∥^{2} ≥ \frac{1}{n}(tr.A)^{2} = \frac{1}{n}(Δf)^{2} in Lemma 4, we arrive at

∫_{M}\{−\frac{(n − 1)}{n}λ∥grad f∥^{2} + Ric(grad f, grad f)\}dv ≤ 0 . (2.7)

As an application of this we have the following corollary which is a well known result [1].

Corollary. Let (M, g) be a compact and connected n- dimensional Riemannian manifold with all sectional curvatures bounded below by a constant k_{0}. Then the first nonzero eigenvalue λ_{1} of the Laplacian satisfies λ_{1} ≥ nk_{0} and the equality holds if and only if M is isometric to a sphere.

Proof. Since all the sectional curvatures are bounded below by k_{0}, we have for a unit vector field X ∈ χ(M) extended to a local orthonormal frame {e_{1}, …, e_{n}} with e_{1} = X, that

Ric(X, X) = ∑_{i} g(R(e_{i}, X)X, e_{i}) ≥ (n − 1)k_{0} .

Thus (2.7) gives (k_{0} − \frac{λ}{n}) ∫_{M} ∥grad f∥^{2}dv ≤ 0, which proves λ ≥ nk_{0}, and

the equality holds if and only if the equality ∥A∥^{2} = \frac{1}{n}(tr.A)^{2} holds, which is

equivalent to A = μI for the function μ = −\frac{λf}{n}. Thus H_{f}(X, Y) = −\frac{−λf}{n} g(X, Y), X, Y ∈ χ(M), where H_{f} is the Hessian of the function f. By Obata’s theorem M is isometric to a sphere [4].

Lemma 5. Let (M, g) be a compact Riemannian manifold and {e_{1}, …, e_{n}} be a local orthonormal frame. Then

∫_{M}{∑_{iα} g(∇_{e_{α}}R)(e_{α}, e_{i})grad f, Ae_{i})}dv

= ∫_{M}\{∑_{iα} R(e_{i}, e_{α}; Ae_{α}, Ae_{i}) − \frac{1}{2} ∑_{iα} ∥R(e_{i}, e_{α})grad f∥^{2}\}dv ,

where R(X, Y; Z, W) = g(R(X, Y)Z, W), X, Y, Z, W ∈ χ(M) and AX = ∇_{X}grad f. Proof. For a local orthonormal frame {e_{1}, …, e_{n}} consider

X = ∑_{i} R(grad f, Ae_{i})e_{i}

and compute

divX = ∑_{α} g(∇_{e_{α}}X, e_{α}) = ∑_{αi} g(∇_{e_{α}}R(grad f, Ae_{i})e_{i}, e_{α})
= ∑_{iα} g((∇_{e_{α}}R)(grad f, Ae_{i})e_{i}, e_{α}) + ∑_{iα} g(R(Ae_{α}, Ae_{i})e_{i}, e_{α})

+ ∑_{iα} g(R(grad f, (∇A)(e_{α}, e_{i}))e_{i}, e_{α})

= ∑_{iα}(∇_{e_{α}}R)(grad f, Ae_{i}; e_{i}, e_{α}) + ∑_{iα} R(Ae_{α}, Ae_{i}; e_{i}, e_{α})

+ ∑_{iα} R(grad f, (∇A)(e_{α}, e_{i}); e_{i}, e_{α}) ,

where R(X, Y; Z, W) = g(R(X, Y)Z, W), X, Y, Z, W ∈ χ(M). This gives

divX = − ∑_{iα}(∇e_{α}R)(e_{α}, e_{i}; grad f, Ae_{i}) + ∑_{iα} R(e_{i}, e_{α}; Ae_{α}, Ae_{i})

+ ∑_{iα} R(e_{i}, e_{α}; grad f, (∇A)(e_{α}, e_{i}))

divX = − ∑_{iα} g((∇e_{α}R)(e_{α}, e_{i}; grad f, Ae_{i}) + ∑_{iα} R(e_{i}, e_{α}; Ae_{α}, Ae_{i})

+ ∑_{iα} g((∇A)(e_{i}, e_{α}) − (∇A)(e_{α}, e_{i}), (∇A)(e_{α}, e_{i})) .

Thus

divX = − ∑_{iα} g((∇e_{α}R)(e_{α}, e_{i})grad f, Ae_{i}) + ∑_{iα} R(e_{i}, e_{α}; Ae_{α}, Ae_{i})

+ ∑_{iα} g((∇A)(e_{i}, e_{α}), (∇A)(e_{α}, e_{i})) − ∥∇A∥^{2} . (2.8)

52 S. DESHMUKH

Next, we have

∑_{iα} ∥R(e_{i}, e_{α})grad f∥^{2}

= ∑_{iα} g((∇A)(e_{i}, e_{α}) − (∇A)(e_{α}, e_{i}), (∇A)(e_{i}, e_{α}) − (∇A)(e_{α}, e_{i}))

= ∑_{iα} ∥(∇A)(e_{i}, e_{α})∥^{2} + ∑_{iα} ∥(∇A)(e_{α}, e_{i})∥^{2} − 2 ∑_{iα} g((∇A)(e_{i}, e_{α}), (∇A)(e_{α}, e_{i}))

= 2∥(∇A)∥^{2} − 2 ∑_{iα} g((∇A)(e_{i}, e_{α}), (∇A)(e_{α}, e_{i})) ,

which gives

−\frac{1}{2} ∑_{iα} ∥R(e_{i}, e_{α})grad f∥^{2} = ∑_{iα} g((∇A)(e_{i}, e_{α}), (∇A)(e_{α}, e_{i})) − ∥∇A∥^{2}

Using above equation in (2.8) and integrating over M we get the result. Lemma 6. Let (M, g) be a compact Riemannian manifold whose Ricci curvature satisfies (∇_{X}Ric)(X, X) = 0, X ∈ χ(M). Then for a local orthonormal frame {e_{1}, …, e_{n}},

∫_{M}{∑_{i} g(∇e_{i}Q(grad f), Ae_{i})}

= \frac{1}{3} ∫_{M}\{∑_{iα} R(e_{i}, e_{α}; Ae_{α}, Ae_{i}) − \frac{1}{2} ∑_{iα} ∥R(e_{i}, e_{α})grad f∥^{2}\}dv

+ ∫_{M}{∑_{i} Ric(Ae_{i}, Ae_{i})}dv .

Proof. The condition (∇_{X}Ric)(X, X) = 0, X ∈ χ(M) is equivalent to

(∇_{X}Ric)(Y, Z) + (∇_{Y} Ric)(Z, X) + (∇_{Z}Ric)(X, Y) = 0, X, Y, Z ∈ χ(M) . (2.9)

Using curvature properties and the second Bianchi identity we obtain

∑_{iα} g((∇_{e_{α}}R)(e_{α}, e_{i})grad f, Ae_{i})

= ∑_{iα}(∇e_{α}R)(e_{α}, e_{i}; grad f, Ae_{i}) = − ∑_{iα}(∇e_{α}R)(grad f, Ae_{i}; e_{i}, e_{α})

= ∑_{iα}(∇_{grad f}R)(Ae_{i}, e_{α}; e_{i}, e_{α}) + ∑_{iα}(∇_{Ae_{i}}R)(e_{α}, grad f; e_{i}, e_{α})

= − ∑_{i}(∇_{grad f}Ric)(Ae_{i}, e_{i}) + ∑_{i}(∇_{Ae_{i}}Ric)(grad f, e_{i}) .

Now use (2.9) in the first term of the above equation to arrive at

(2.10) ∑_{iα} g((∇_{e_{α}}R)(e_{α}, e_{i})grad f, Ae_{i})

= 2 ∑_{i}(∇_{Ae_{i}}Ric)(grad f, e_{i}) + ∑_{i}(∇_{e_{i}}Ric)(grad f, Ae_{i}) .

Since (∇_{X}Ric)(Y, Z) = g((∇_{X}Q)(Y), Z) and Ae_{i} = ∑_{k} g(Ae_{i}, e_{k})e_{k}, we get

∑_{i}(∇_{Ae_{i}}Ric)(grad f, e_{i})

= ∑_{ik} g(Ae_{i}, e_{k})(∇_{e_{k}}Ric)(grad f, e_{i}) = ∑_{ik} g(Ae_{i}, e_{k})g((∇_{e_{k}}Q)(grad f), e_{i})

= ∑_{k} g((∇_{e_{k}}Q)(grad f), ∑_{i} g(Ae_{k}, e_{i})e_{i}) = ∑_{k} g((∇_{e_{k}}Q)(grad f), Ae_{k})

= ∑_{k}(∇_{e_{k}}Ric)(grad f, Ae_{k}) .

Thus (2.10) becomes

\frac{1}{3} ∑_{iα} g((∇_{e_{α}}R)(e_{α}, e_{i})grad f, Ae_{i})

= ∑_{i} g((∇_{e_{i}}Q)(grad f), Ae_{i}) = ∑_{i} g(∇_{e_{i}}Q(grad f), Ae_{i})

− ∑_{i} g(Q(Ae_{i}), Ae_{i}) = ∑_{i} g(∇_{e_{i}}Q(grad f), Ae_{i}) − ∑_{i} Ric(Ae_{i}, Ae_{i}) .

Integrating this last equation and using Lemma 5, we get the result. Lemma 7. Let (M, g) be an n-dimensional connected Riemannian manifold. Then

∥∇A∥^{2} ≥ \frac{1}{n}∥gradΔf∥^{2}

and for a positively curved M, the equality holds if and only if A(X) = (−\frac{Δf}{n} ) X, X ∈ χ(M), where AX = ∇_{X}grad f, X ∈ χ(M).

Proof. Define B(X, Y) = g(AX, Y)+ \frac{Δf}{n} g(X, Y), X, Y ∈ χ(M). Then we obtain

(∇B)(X, Y, Z) = g((∇A)(X, Y), Z) + \frac{X(Δf)}{n}g(Y, Z), X, Y, Z ∈ χ(M).

Thus

∥∇B∥^{2} = ∑_{ijk}[(∇B)(e_{i}, e_{j}, e_{k})]^{2}

= ∥∇A∥^{2} + \frac{1}{n}∥gradΔf∥^{2} + \frac{2}{n} ∑_{ij} g((∇A)(e_{i}, e_{j}), e_{j})g(gradΔf, e_{i}) .

Now use equation (2.2) to obtain

∥∇B∥^{2} = ∥∇A∥^{2} + \frac{1}{n}∥gradΔf∥^{2}

+ \frac{2}{n} ∑_{ij} g((∇A)(e_{j}, e_{i}) + R(e_{i}, e_{j})grad f, e_{j})g(gradΔf, e_{i})

= ∥∇A∥^{2} + \frac{1}{n}∥gradΔf∥^{2} + \frac{2}{n} ∑_{ij}[g(e_{i}, (∇A)(e_{j}, e_{j}))g(gradΔf, e_{i})

− Ric(e_{i}, grad f)g(gradΔf, e_{i})] .

Using Lemma 2, we find

∥∇B∥^{2} = ∥∇A∥^{2} + \frac{1}{n}∥gradΔf∥^{2} − \frac{2}{n}∥gradΔf∥^{2} = ∥∇A∥^{2} − \frac{1}{n}∥gradΔf∥^{2} ,

which proves ∥∇A∥^{2} ≥ \frac{1}{n}∥gradΔf∥^{2}. If the equality holds, we should have ∇B = 0, and as M is irreducible (M being of positive curvature), we get B(X, Y) = ag(X, Y) for some constant a ∈ R. This together with the definition of B gives

A(X) = (a − \frac{Δf}{n} ) (X). Since tr.A = −Δf, we get −Δf = (a − \frac{Δf}{n} ) n, which

implies a = 0; that is, A(X) = (−\frac{Δf}{n} ) X, X ∈ χ(M).

Lemma 8. Let (M, g) be a compact Riemannian manifold and {e_{1}, …, e_{n}} be a local orthonormal frame which diagonalizes A with Ae_{i} = λ_{i}e_{i}. Then

∫_{M}{∑_{i<α}(λ_{i} − λ_{α})^{2}}dv = (n − 1) ∫_{M}(Δf)^{2}dv − n ∫_{M} Ric(grad f, grad f)dv ,

where A(X) = ∇_{X}grad f, X ∈ χ(M).

Proof. We have ∑_{iα}(λ_{i} − λ_{α})^{2} = ∑_{iα} λ_{i}^{2} + ∑_{iα} λ_{α}^{2} − 2 ∑_{iα} λ_{i}λ_{α} = n∥A∥^{2} + n∥A∥^{2} −

2 ∑_{α}(−Δf)λ_{α} = 2n∥A∥^{2} − 2(Δf)^{2} and ∑_{iα}(λ_{i} − λ_{α}^{2}) = 2 ∑_{i<α}(λ_{i} − λ_{α})^{2}.

Integrating the resulting equation and using Lemma 4, we get the result.

3. Proof of the Theorem

Define F : M → R by F = \frac{1}{2}∥A∥^{2}. Then it is straight-forward to see that the Hessian H_{F} of F is given by

H_{F} (X, Y) = ∑_{ij} g((∇A)(X, e_{i}), e_{j})g((∇A)(Y, e_{i}), e_{j})

+ ∑_{ij} g(Ae_{i}, e_{j})g((∇^{2}A)(X, Y, e_{i}), e_{j}) ,

CURVATURE BOUNDS FOR THE SPECTRUM … 55

which gives

H_{F} (X, X) = ∑_{ij} g((∇A)(X, e_{i}), e_{j})^{2} + ∑_{i} g((∇^{2}A)(X, X, e_{i}), Ae_{i}) .

Now, using equation (2.2) and Lemma 1 in the second term of the above equation, we arrive after some simple calculations, at

H_{F} (X, X) = ∑_{ij} g((∇A)(X, e_{i}), e_{j})^{2}

+ ∑_{i}[g((∇^{2}A)(e_{i}, X, X), Ae_{i}) + 2R(X, e_{i}; AX, Ae_{i})

− R(X, e_{i}; X, A^{2}e_{i}) + g((∇_{X}R)(X, e_{i})grad f, Ae_{i})] .

Using −ΔF = ∑_{α} H_{F} (e_{α}, e_{α}) in this equation we get

(3.1) − ΔF = ∥∇A∥^{2} + ∑_{iα}[g((∇^{2}A)(e_{i}, e_{α}, e_{α}), Ae_{i}) + 2R(e_{α}, e_{i}; Ae_{α}, Ae_{i})

− R(e_{α}, e_{i}; e_{α}, A^{2}e_{i}) + g((∇_{e_{α}}R)(e_{α}, e_{i})grad f, Ae_{i})]

From Lemma 2, we find

∑_{α}(∇^{2}A)(X, e_{α}, e_{α}) = −∇_{X}gradΔf + ∇_{X}Q(grad f) ,

which, together with (3.1), gives

(3.2) − ΔF = ∥∇A∥^{2} + ∑_{i}[−g(∇_{e_{i}}gradΔf, Ae_{i}) + g(∇_{e_{i}}Q(grad f), Ae_{i})]

+ ∑_{iα}[2R(e_{α}, e_{i}; Ae_{α}, Ae_{i}) − R(e_{α}, e_{i}; e_{α}, A^{2}e_{i})]

+ ∑_{iα} g((∇_{e_{α}}R)(e_{α}, e_{i})grad f, Ae_{i}) .

Next, we compute

(3.3) div(A(gradΔf)) = ∑_{i} g((∇A)(e_{i}, gradΔf) + A(∇_{e_{i}}gradΔf), e_{i})

= ∑_{i} g(gradΔf, (∇A)(e_{i}, e_{i})) + ∑_{i} g(∇_{e_{i}}gradΔf, Ae_{i})

= −∥gradΔf∥^{2} + Ric(grad f, gradΔf) + ∑_{i} g(∇_{e_{i}}gradΔf, Ae_{i}) .

56 S. DESHMUKH

Substituting the value of ∑_{i} g(∇_{e_{i}}gradΔf, Ae_{i}) as expressed by (3.3) into (3.2), integrating the resulting equation and using Lemmas 5 and 6, we get

∫_{M}{∥∇A∥^{2} − ∥gradΔf∥^{2} + Ric(grad f, gradΔf)

+ \frac{1}{3} ∑_{iα}[4R(e_{i}, e_{α}; Ae_{α}, Ae_{i}) − 2∥R(e_{i}, e_{α})grad f∥^{2}]

+ ∑_{i} Ric(Ae_{i}, Ae_{i}) + ∑_{iα}[2R(e_{α}, e_{i}, Ae_{α}, Ae_{i})

− R(e_{α}, e_{i}, e_{α}, A^{2}e_{i})]}dv = 0 .

Taking Ae_{j} = λ_{j}e_{j} in the above integral and noting that

∑_{i} Ric(Ae_{i}, Ae_{i}) − ∑_{iα} R(e_{i}, e_{α}; Ae_{α}, Ae_{i})

= ∑_{iα} λ_{i}^{2}K_{iα} − ∑_{iα} λ_{i}λ_{α}K_{iα} = \frac{1}{2}[∑_{iα} λ_{i}^{2}K_{iα} + ∑_{iα} λ_{α}^{2}K_{iα} − 2 ∑_{iα} λ_{i}λ_{α}K_{iα}]

= \frac{1}{2} ∑_{iα}(λ_{i} − λ_{α})^{2}K_{iα} = ∑_{i<α}(λ_{i} − λ_{α})^{2}K_{iα} ,

where K_{iα} = R(e_{i}, e_{α}; e_{α}, e_{i}) is the sectional curvature of the plane section spanned by {e_{i}, e_{α}}, we get (3.4) ∫_{M}{∥∇A∥^{2} − ∥gradΔf∥^{2} + Ric(grad f, gradΔf)

+ \frac{2}{3} ∑_{i<α}(λ_{i} − λ_{α})^{2}K_{iα} + \frac{4}{3} ∑_{i} Ric(Ae_{i}, Ae_{i}) − \frac{2}{3} ∑_{iα} ∥R(e_{i}, e_{α})gradf∥^{2}\}dv = 0 .

Since all the sectional curvatures are bounded below by k_{0}, we have

∫_{M}{∑_{i} Ric(Ae_{i}, Ae_{i})}dv ≥ k_{0}(n − 1) ∫_{M} ∥A∥^{2}dv
= k_{0}(n − 1) ∫_{M}{(Δf)^{2} − Ric(grad f, grad f)}dv

where we used Lemma 4. Now, using Lemma 8, we obtain

∫_{M}{∑_{i} Ric(Ae_{i}, Ae_{i})}dv ≥ k_{0}(n − 1) ∫_{M}(Δf)^{2}dv

−\frac{k_{0}(n − 1)}{n} ∫_{M}\{(n − 1)(Δf)^{2} − ∑_{i<α}(λ_{i} − λ_{k})^{2}\}dv

CURVATURE BOUNDS FOR THE SPECTRUM … 57

that is,

(3.5) ∫_{M}\{∑_{i} Ric(Ae_{i}, Ae_{i})\}dv ≥ k_{0} \frac{(n − 1)}{n} ∫_{M}(Δf)^{2}dv

+k_{0} \frac{(n − 1)}{n} ∫_{M}\{∑_{i<α}(λ_{i} − λ_{α})^{2}\}dv .

Now suppose that Δf = λf, λ > 0, then we have

∫_{M} Ric(grad f, gradΔf)dv = λ ∫_{M} Ric(grad f, grad f)dv

= \frac{λ}{n} ∫_{M} nRic(grad f, grad f)dv ,

which in view of Lemma 8, gives (3.6) ∫_{M} Ric(grad f, gradΔf)dv

= \frac{λ(n − 1)}{n} ∫_{M}(Δf)^{2}dv − \frac{λ}{n} ∫_{M}\{∑_{i<α}(λ_{i} − λ_{α})^{2}\}dv .

Substituting from (3.5) and (3.6) into (3.4) and using the relations

∑_{iα} ∥R(e_{i}, e_{α})grad f∥^{2} = ∥R_{grad f}∥^{2} ≤ 2k_{0}^{2}(n − 1)L∥grad f∥^{2}, and

∫_{M}(Δf)^{2}dv = λ ∫_{M} ∥grad f∥^{2}dv which is an outcome of Lemma 3, we obtain

(3.7) ∫_{M}(∥∇A∥^{2} − \frac{1}{n}∥grad∇f∥^{2})dv

+ \frac{2}{3n} ((3n − 2)k_{0} − \frac{3λ}{2}) ∫_{M}\{∑_{i<α}(λ_{i} − λ_{α})^{2}\}dv

+ \frac{4}{3}k_{0}(n − 1)[\frac{λ}{n} − Lk_{0}] ∫ ∥grad f∥^{2}dv ≤ 0 .

If both \frac{λ}{n} − Lk_{0} ≥ 0 and (3n − 2)k_{0} − \frac{3λ}{2} ≥ 0, then we must have by Lemma 7, ∥∇A∥^{2} = \frac{1}{n}∥gradΔf∥^{2} and λ = nLk_{0} > 0,(as f is a non-constant function Δf = λf, λ > 0), which again by Lemma 7 implies that A(X) = −\frac{λf}{n} X, that is, H_{f}(X, Y) = \frac{−λf}{n} g(X, Y). Hence by Obata’s theorem M is isometric to a sphere. Otherwise we have one of the following (i) (3n − 2)k_{0} − \frac{3λ}{2} < 0, (ii) \frac{λ}{n} − Lk_{0} < 0 (iii) (3n − 2)k_{0} − \frac{3λ}{2} < 0 and \frac{λ}{n} − Lk_{0} < 0, and this proves the theorem.

4. Characterization of spheres

In this section we shall use the length ∥R_{X}∥ of the curvature tensor field in the direction of a vector field X to obtain a characterization of the spheres. Note that for a sphere S^{n}(k_{0}) we have ∥R_{X}∥^{2} = 2(n − 1)k_{0}^{2} ∥X∥^{2} and the first nonzero eigenvalue λ_{1} is given by λ_{1} = nk_{0}. Thus we have

∥R_{X}∥^{2} = \frac{(n − 1)}{2n}λ_{1}(4k_{0}) ∥X∥^{2} = \frac{n − 1}{2n}λ_{1} [(3n + 4)k_{0} − 3λ_{1}] ∥X∥^{2} .

This raises a question, “Is a compact and connected positively curved n-dimensional Riemannian manifold satisfying

∥R_{X}∥^{2} = \frac{(n − 1)}{2n}λ_{1} [(3n + 4)k_{0} − 3λ_{1}] ∥X∥^{2} ,

where k_{0} is the infimum of the sectional curvatures, isometric to S^{n}?”

We show that the answer to this question is in affirmative for the Riemannian manifolds of class A. In fact we prove the following:

Theorem 4.1. Let (M, g) be an n-dimensional compact and connected Riemannian manifold whose Ricci curvature satisfies

(∇_{X}Ric)(X, X) = 0, X ∈ χ(M) ,

and all its sectional curvatures be bounded below by a constant k_{0} > 0. If the first nonzero eigenvalue λ_{1} of the Laplacian satisfies

∥R_{X}∥^{2} ≤ \frac{(n − 1)}{2n}λ_{1} [(3n + 4)k_{0} − 3λ_{1}] ∥X∥^{2} , X ∈ χ(M) ,

where ∥R_{X}∥^{2} = ∑_{ij} ∥R(e_{i}, e_{j})X∥^{2} , then M is isometric to S^{n}(k_{0}).

Proof. Let f : M → R be a smooth function such that Δf = λf, λ > 0. Then equation (3.4) takes the form

∫_{M} \{∥∇A∥^{2} − \frac{1}{n} ∥gradΔf∥^{2}\} dv

+ ∫_{M} \{(−\frac{(n − 1)}{n}λ^{2} + (n − 1)k_{0}λ) ∥grad f∥^{2} − \frac{2}{3} ∥R_{grad f}∥^{2}\} dv

+ \frac{4}{3} ∫_{M} \{∑_{i} Ric(Ae_{i}, Ae_{i})\} dv + \frac{2}{3}k_{0} ∫_{M} \{∑_{i<j}(λ_{i} − λ_{j})^{2}\} dv ≤ 0

Using Lemma 3 in §2, in (3.5) we arrive at

∫_{M} {∑_{i} Ric(Ae_{i}, Ae_{i})} dv

≥ k_{0} \frac{(n − 1)}{n}λ ∫_{M} ∥grad f∥^{2} dv + \frac{k_{0}(n − 1)}{n} ∫_{M} \{∑_{i<j}(λ_{i} − λ_{j})^{2}\} dv .

Thus (4.1) takes the form

∫_{M} \{∥∇A∥^{2} − \frac{1}{n} ∥gradΔf∥^{2}\} dv

+ \frac{2}{3} ∫_{M} \{λ\frac{(n − 1)}{2n} [(3n + 4)k_{0} − 3λ] − ∥R_{grad f}∥^{2}\} dv

+ \frac{2}{3} \frac{(3n − 2)}{n}k_{0} ∫_{M} \{∑_{i<j}(λ_{i} − λ_{j})^{2}\} dv ≤ 0 .

If the given condition holds, as the integrand in first integral is non-negative, we have from the last integral A = μI, which gives μ = −\frac{λ_{1}f}{n}, consequently H_{f}(X, Y) = −\frac{λ_{1}f}{n} g(X, Y) holds. Hence by Obata’s theorem M is isometric to the sphere S^{n}(c) whose ∥R_{grad f}∥ according to (4.2) is given by

∥R_{grad f}∥^{2} = \frac{λ_{1}(n − 1)}{2n} [(3n + 4)k_{0} − 3λ_{1}] ∥grad f∥^{2}

= \frac{(n − 1)}{2}c [(3n + 4)k_{0} − 3nc] ∥grad f∥^{2} ,

as λ_{1} = nc. However ∥R_{grad f}∥ for S^{n}(c) is ∥R_{grad f}∥^{2} = 2(n − 1)c^{2} ∥grad f∥^{2} using this in (4.3), we get c = k_{0} and consequently M is isometric to S^{n}(k_{0}).

Similarly for Einstein manifolds we prove the following

Theorem 4.2. Let (M, g) be an n-dimensional compact and connected nonnegatively curved Einstein manifold of constant positive scalar curvature S. If the length ∥R_{X}∥ and the first nonzero eigenvalue λ_{1} of the Laplacian satisfy

∥R_{X}∥^{2} ≤ \frac{(n − 1)}{2n}λ_{1} [\frac{(3n + 4)S}{n(n − 1)} − 3λ_{1}] ∥X∥^{2} , X ∈ χ(M) ,

then M is isometric to S^{n} (\frac{S}{n(n−1)}) .

Proof. Since, Ric = \frac{S}{n} g, for a smooth f : M → R with Δf = λf, λ > 0, the equation (3.4) gives

∫_{M} \{∥∇A∥^{2} − \frac{1}{n} ∥gradΔf∥^{2}\} dv

+∫_{M} \{−\frac{(n − 1)}{n}λ^{2} ∥grad f∥^{2} + \frac{S}{n} λ ∥grad f∥^{2} + \frac{4}{3} ∥A∥^{2} − \frac{2}{3} ∥R_{grad f}∥^{2}\} dv

+ ∫_{M} {∑_{i<j}(λ_{i} − λ_{j})^{2}K_{ij}} dv = 0 .

Now, using lemmas 4 and 8, we get

∫_{M} \{∥∇A∥^{2} − \frac{1}{n} ∥gradΔf∥^{2}\} dv

+ ∫_{M} \{\frac{λ}{n} [\frac{S(3n + 4)}{3n} − (n − 1)λ] ∥grad f∥^{2} − \frac{2}{3} ∥R_{grad f}∥^{2}\} dv

+ ∫_{M} \{∑_{i<j}(λ_{i} − λ_{j})^{2}(K_{ij} + \frac{4}{3} \frac{S}{n^{2}} )\} dv = 0 .

If the given condition holds, then as in the proof of last theorem, from above equation we conclude that M is isometric to S^{n}(c) and it can be similarly shown that c = \frac{S}{n(n−1)}.

References

[1] Chavel, I., Eigenvalues in Riemannian Geometry, Academic Press (1984).

[2] Gallot, S., Hulin, D. and Lafontaine, J., Riemannian Geometry, Springer Verlag (1990).

[3] Gray, A., Einstein-like manifolds which are not Einstein, Geom. Ded. 7(1978), 259–280.

[4] Obata, M., Certain conditions for a Riemannian manifold to be isometric with a sphere, J. Math. Soc. Japan 14(1962), 333–340.

[5] Simon, U., Curvature bounds for the spectrum of a closed Einstein spaces, Can. J. Math. XXX(4)(1978), 1087–1091.

[6] Tanno, S., Eigenvalues of the Laplacian of Riemannian manifolds, Tohoku Math. J. 25(1973), 391–403.

Department of Mathematics, King Saud University P.O. Box 2455 Riyadh 11451, SAUDI ARABIA E-mail: shariefd@ksu.edu.sa