ON THE PROJECTIONS OF LAPLACIANS UNDER RIEMANNIAN SUBMERSIONS

We give a condition on Riemannian submersions from a Riemannian manifold M to a Riemannian manifold N which will ensure that it induces a differential operator on N from the Laplace-Beltrami operator on M . Equivalently, this condition ensures that a Riemannian submersion maps Brownian motion to a diffusion. 2000 Mathematics Subject Classification. Primary 58J65, 53C21.

1. Introduction.Suppose that M, N are, respectively, m-and n-dimensional Riemannian manifolds and that m > n.Both M and N will then carry Laplace-Beltrami operators M and N , respectively, determined by the Riemannian metrics.
Let the mapping π : M → N such that π(σ m ) = σ n be a Riemannian submersion.Normally, the Laplace-Beltrami operator M will not induce a differential operator on N under the submersion π because M may depend not only on π(σ m ) but also on σ m .Equivalently, a Brownian motion on M will not normally be mapped by π to a diffusion on N because it may happen that our prediction of σ n (t + u) (u > 0) will be improved if we know where σ m (t) lies in π −1 (σ n (t)), and we can expect to get information about σ n (t) from the past history {σ n (t − v) : 0 ≤ v < t} of the submersed process.However, once we know that there is a differential operator ᏸ on N that satisfies the relation we can find several equivalent expressions for ᏸ in terms of the volume, the second fundamental form, and the mean curvature of the fibres, respectively, which will be listed here.
(a) If the fibres are compact, let v(σ n ) be the (m − n)-dimensional volume of the fibre π −1 (σ n ) and V the vector field grad(log v).Carne's formula (cf.[3]) then tells us that Recall that M can be written in terms of any given orthonormal vector fields X 1 ,...,X m on M as the operator ∇ here being the Levi-Civita connection.Therefore, we choose Y 1 ,...,Y n to be orthonormal vector fields in a neighborhood of σ n ∈ N, X 1 ,...,X n the unique horizontal lifts of Y 1 ,...,Y n to a neighborhood of σ m ∈ π −1 (σ n ) (so that X 1 ,...,X n are orthonormal vector fields on the π -related horizontal subspace of -(M)) and then supplement the latter by m − n orthonormal vertical vector fields X n+1 ,...,X m in the same neighborhood.M at σ m can thus be written as However, for any smooth function φ : N → R, the composed function φ • π : M → R will be constant along each fibre π −1 (σ n ), and hence And, on the other hand, ∇ X i X i is equal to the sum of the horizontal lift of The Hessian of a function φ is the symmetric (0, 2) tensor field defined by and the so-called shape tensor (or "second fundamental form" tensor) of each fibre π −1 (σ n ) is the bilinear symmetric mapping Π from , where ᐄ(π −1 (σ n )) denotes the set of all smooth vertical vector fields of M defined on and so an equivalent expression for ᏸ is (1.9) (c) Moreover, for any (m−n)-dimensional submanifold M 0 of M, the mean curvature vector field H M 0 of M 0 at p ∈ M 0 is given by where E n+1 ,...,E m is any orthonormal basis for the tangent spacep (M 0 ).It is easy to check that if x n+1 ,...,x m is an adapted coordinate system for M 0 , then .11)and that if ∂/∂x 1 ,...,∂/∂x n are normal to M 0 , then (1.12) It follows from (1.9) that which gives another expression for ᏸ when it exists.So a problem rises here: what is the condition for such a differential operator ᏸ to exist, that is, when does the submersion π map a Brownian motion on M to a diffusion on N?
The above discussion shows that M (φ • π) = (ᏸφ) • π for some operator ᏸ on N if and only if the traces of the second fundamental form for each fibre π −1 (σ n ) are π -related on that fibre; or equivalently, if and only if the mean curvature vector fields H π −1 of each fibre π −1 (σ n ) are π -related on that fibre, for evidently either of these is the necessary and sufficient condition that M depends only on π(σ m ), and not on σ m itself.
We now discuss another condition in terms of the volume element of M for the existence of ᏸ.

Some lemmas
Lemma 2.1.Let G M and G N be the matrices of the local components of the metric tensor fields on M and N with respect to local coordinates x : σ m → (x 1 ,...,x m ) on M and y : σ n → (y 1 ,...,y n ) on N, respectively, then where J is the Jacobian matrix of the coordinate representation y and J t is its transpose.
For any given local coordinate y on N at σ n , there exists a local coordinate This implies that π is locally a fibration, that is, there exist a neighborhood U n of is complete.It follows that there are m − n independent solutions x n+1 ,...,x m of (2.4), such that general solution of (2.4) is an arbitrary function of x n+1 ,...,x m (cf.[4]).Define is the coordinate we are looking for.In fact, for any given coordinate y in N we can always find a coordinate x in M such that (2.3) holds.Each X i can then be formulated as where if the metric form of M with respect to x is (2.7) Thus, the metric form of M with respect to x, by the fact that X i x j = 0, for 1 ≤ i ≤ n < j ≤ m, will be for some positive definite symmetric matrix H.

Main result
Proposition 3.1.If π is integrable, then there is an operator ᏸ on N with if and only if the volume element dµ M of M can be expressed as a product of two independent forms: one is a composed n-form on N with the submersion π defined by and the other is an (m − n)-form on the fibres π −1 (σ n ), the local expression of which is denoted by with the property that the latter will be independent of σ n in a neighborhood of σ n .And when this condition is satisfied, Proof.The local form of the Laplace-Beltrami operator, in terms of any given coordinate x on M, is Thus for the coordinates x and y as Lemma 2.2, we are able to obtain that, for any smooth function φ : N → R, Note that here ∂/∂x 1 ,...,∂/∂x n are the horizontal lifts of ∂/∂y 1 ,...,∂/∂y n .We know from the assumption that is a distribution, and forms a basis for it.Following the same discussion as in the proof of Lemma 2.2, we know that any solution of the system of differential equations is a function of x n+1 ,...,x m .On the other hand, we have by (1.2) that existence of ᏸ on N if and only if there is a function Φ on N such that Therefore, the existence of ᏸ is equivalent to that there is a function e Φ det G N is clearly a function on N. If we define a function Ψ * on a neighborhood of σ m ∈ π −1 (σ n ), as the restriction of the function e (1/2)Ψ on π −1 (σ n ), then Ψ * is independent on fibres in a neighborhood of σ m ∈ π −1 (σ n ), and det G M is a product of a composed function on N with π and a function on the fibres of π .
The above discussion shows that the volume element dµ M on M is here expressed as (3.12) Because π is a submersion, M is locally diffeomorphic to N × F for a (m − n)dimensional manifold F , and so the above condition is equivalent to that the volume element dµ M can locally be expressed as a product of a composed n-form on N with the submersion π and an (m − n)-form on F .

Remarks. (a)
We know from the proof of Proposition 3.1 that, for any general coordinates such that (2.3) holds, Compared with (1.6), we know that the kth And compared with (1.2), we find that there is a differential operator ᏸ on N with (ᏸφ)•π = M (φ•π) if and only if, for any 1 ≤ k ≤ n, (4.3) is a function of π(σ m ), and where that is, the first n components of grad M {(1/2) log(volume element of the fibre π −1 (σ n ))} do not form a proper gradient of a function on N, which usually depend not only on π(σ m ) but also on σ m .When Equation (4.4) can be rewritten as is orthogonal with all dx k for 1 ≤ k ≤ n in - * (M).(b) When the condition in Proposition 3.1 holds, the volume element of the fibre and so if π is also a fibration with compact fibre F , the (m − n)-dimensional volume v(σ n ) of the fibre π −1 (σ n ) will then be equal to for some constant κ, which coincides with (1.2).(c) The condition of integrability of π in Proposition 3.1 should be able to be weakened.We study the following two cases.
(i) For the submersion π with minimal fibres, in particular with totally geodesic fibres, it is known that ᏸ = N , which follows immediately from the fact that the term in (1.6) vanishes by the definition of minimal submanifold.On the other hand, when M is complete and π with totally geodesic fibres, we can also obtain from the fact that (M,N,π) is a fibre bundle with the Lie group of isometries of the fibre as structure group (cf.[5] and below) that for a suitable coordinate (x n+1 ,...,x m ) on fibres.
In the case that π is with minimal fibres, it follows from the fact that the structure group of the bundle (which is a priori the group of diffeomorphisms of the fibre F ) reduces to the group of volume preserving diffeomorphisms of F (cf. [1]) that the volume element of M is of the expression (4.15).
(ii) The case that the submersion π is a quotient mapping with respect to a Lie group G of isometries acting properly and freely on M.
The fibre π −1 (σ n ) here inherits a Riemannian structure from that of M, and the corresponding volume element dµ π −1 (σn) of the fibre π −1 (σ n ) is invariant under G by the transitive action of G of isometries on the fibres.Under the identification π −1 (σ n ) = G, the volume elements dµ π −1 (σn) and dg, the unique left-invariant volume element up to constants of G, must, by the uniqueness, be proportional (cf.[2]).Hence there exists a function e (1/2)Φ on N such that dµ π −1 (σn) = e (1/2)Φ(σn) dg, (4.16) and so dµ M = dg e (1/2)Φ dµ N • π, (4.17) which gives a form for the volume element on M coincident with our claim if we notice that here M is locally diffeomorphic to N × G.