JeongHyeong Park

UNIVERSAL CURVATURE IDENTITIES III

We examine universal curvature identities for pseudo-Riemannian manifolds with boundary. We determine the Euler–Lagrange equations associated to the Chern–Gauss–Bonnet formula and show that they are given solely in terms of curvature and the second fundamental form and do not involve covariant derivatives, thus generalizing a conjecture of Berger to this context.

Eigenvalues of the form valued Laplacian for Riemannian submersions

Let π : Z → Y be a Riemannian submersion of closed manifolds. Let Φp be an eigen p-form of the Laplacian on Y with eigenvalue λ which pulls back to an eigen p-form of the Laplacian on Z with eigenvalue μ. We are interested in when the eigenvalue can change. We show that λ ≤ μ, so the eigenvalue can only increase; and we give some examples where λ < μ, so the eigenvalue changes. If the horizontal distribution is integrable and if Y is simply connected, then λ = μ, so the eigenvalue does not change. If M is a closed Riemannian manifold, let E(λ,∆pM ) be the eigenspace of the Laplacian ∆pM := dδ + δd for the eigenvalue λ on the space of smooth p-forms C∞(ΛpM). Let π : Z → Y be a Riemannian submersion of closed manifolds. Pullback defines a natural map π∗ from C∞(ΛpY ) to C∞(ΛpZ). We are interested in examples where an eigenform on Y pulls back to an eigenform on Z with a different eigenvalue. Let V and H be the vertical and horizontal distributions of π. We say that H is an integrable SL (Special Linear) distribution if H is integrable and if there exists a measure ν on the fibers of π so that the Lie derivative LHν vanishes for all horizontal lifts H ; this means that the transition functions of the fibration can be chosen to have Jacobian determinant 1. Theorem 1. Let π : Z → Y be a Riemannian submersion of closed manifolds. Let Φp ∈ E(λ,∆pY ) and π∗Φp ∈ E(μ,∆pZ). If H is an integrable SL distribution, λ = μ. We will show in Lemma 6 that if Y is simply connected and if H is integrable, then H is an integrable SL distribution and therefore eigenvalues do not change. This shows the fundamental role that the curvature tensor plays in this subject. We note that the Godbillon-Vey class of the foliation H vanishes if H is an integrable SL distribution. In the general setting, we show that if eigenvalues change, they can only increase. Theorem 2. Let π : Z → Y be a Riemannian submersion of closed manifolds. Let Φp ∈ E(λ,∆pY ) and π∗Φp ∈ E(μ,∆pZ). Then λ ≤ μ. It is not difficult to use the maximum principle to show that eigenvalues cannot change if p = 0. It is not known if eigenvalues can change if p = 1. Eigenvalues can change if p ≥ 2. Received by the editors May 20, 1996. 1991 Mathematics Subject Classification. Primary 58G25.

INVARIANT METRICS OF POSITIVE RICCI CURVATURE ON PRINCIPAL BUNDLES

Abstract. Let

A generalization of a 4-dimensional Einstein manifold

Y

The spectral geometry of the Hopf fibration

be a compact connected Riemannian manifold with a metric of positive Ricci curvature. Let

Heat trace asymptotics of a time dependent process

\pi:P\rightarrow Y

NOTES ON CRITICAL ALMOST HERMITIAN STRUCTURES

be a principal bundle over

Asymptotics of the heat equation with ‘exotic’ boundary conditions or with time dependent coefficients

Y

Heat content asymptotics

with compact connected structure group

Eigenforms of the Laplacian for Riemannian V-submersions

G