(1) the sets U± cover M,

’1

’1

(2) for any U± , Uβ with U± © Uβ = … the functions •β —¦ •± and •± —¦ •β are in C k on •± (U± © Uβ )

and •β (U± © Uβ ), respectively.

Finally, a manifold M is called a C k -manifold if it possesses a C k -atlas.

The smoothness of a function f : M ’ R is de¬ned by the smoothness of f —¦ • ’1 with

a chart (U, •). To be more precise, we will say that f is k times differentiable on M, or

f ∈ C k (M), if f —¦ • ’1 ∈ C k (•(U )) for every chart (U, •) of M. In what follows we will

assume that the underlying manifold is suf¬ciently smooth.

Most relevant examples such as the sphere and the torus are submanifolds of Rd . This

means in particular that they inherit the standard metric of Rd , which is induced by the

Euclidean norm. However, everything we have in mind works in the more general setting

of Riemannian manifolds. To introduce them, we have to recall concepts regarding curves

on manifolds and tangent spaces.

For x ∈ M, the tangent space Tx (M) consists of all tangent vectors v to M in x. Here,

a vector v is a tangent vector if there exists a differentiable curve γ : [’ , ] ’ M with

γ (0) = x and γ (0) = v. It turns out that Tx (M) is an n-dimensional subspace of Rd and

that a basis is given by

‚• ’1 ‚• ’1

...,

(•(x)), (•(x)).

‚v1 ‚vn

De¬nition 17.16 A C k -manifold is called a C k Riemannian manifold if for every x ∈ M

there exists an inner product gx : Tx (M) — Tx (M) ’ R such that for every coordinate

neighborhood (U, •) the n 2 functions

‚• ’1 ‚• ’1

•

(v) , v ∈ V = •(U ),

gi j (v) := g• ’1 (v) (v),

‚vi ‚v j

are in C k (V ).

Finally, we use the Riemannian metric to de¬ne the length of a curve on M.

De¬nition 17.17 Suppose that M is a C k Riemannian manifold. Let x, y ∈ M be two

distinct points and let γ : [a, b] ’ M be a piecewise C 1 curve that connects these points,

i.e. γ (a) = x, γ (b) = y. Then the length of γ is

1/2

b

dγ dγ

L(γ ) = L(γ ; a, b) := gγ (t) (t), (t) dt.

dt dt

a

Finally, we set dist(x, y) to be the in¬mum over the length of all such curves connecting x

and y. The shortest such curve is called the shortest path for x and y, and dist(x, y) is their

geodesic or Riemannian distance.

318 Interpolation on spheres and other manifolds

Several remarks are necessary. First of all, if M = Rn and if gx is the canonical inner

product on Rn then our de¬nition of the length of a curve coincides with the classical

de¬nition. In this case dist(x, y) = x ’ y 2 , i.e the shortest curve between two points

in Rn is the straight line between them. In general, dist de¬nes a metric on M, if M is

connected. The latter assumption is obviously necessary to make dist always well de¬ned.

Even more, the topology induced by dist is equivalent to the initial topology on M. We will

come back to this later. On the sphere, our new de¬nition of dist coincides with the old one,

since both denote the length of the shorter portion of the great circle connecting the two

points.

The inner products gx , x ∈ M, are necessary for introducing the concept of integration

on M. Suppose M is a compact C k Riemannian manifold with C k -atlas A = {(U j , • j )} L .

j=1

For this atlas one can choose a partition of unity, i.e. a family of functions {χ j } such

χ j = 1 on M and supp(χ j ) ⊆ U j . Moreover, a reasonable choice makes χ j —¦ • ’1

that j

integrable over • j (U ). Then the integral for a measurable function f on M is de¬ned by

L

(χ j f ) —¦ • ’1 (v)g j (v)dv,

1/2

f (x)dS(x) = j

• j (U j )

M j=1

•

with g j (v) := det(gikj (v)). Of course, one has to show that the integral is independent of the

chosen atlas and the chosen partition of unity. In this sense, spaces of integrable functions

can be introduced.

After reviewing the basic concepts of Riemannian manifolds, we return to the study

of positive de¬nite kernels on them. A ¬rst obvious but also intrinsic observation is the

following.

Proposition 17.18 Suppose that M is a differentiable manifold and (U, •) is a chart. If

: M — M ’ R is a positive de¬nite kernel on M then

(• ’1 (u), • ’1 (v)),

(u, v) := u, v ∈ •(U ),

is a positive de¬nite kernel on •(U ).

Reversing the argument, one could use this result to prove the positive de¬niteness of a

kernel de¬ned on M. But since everything depends on the charts that are chosen, the use of

Proposition 17.18 in this context is restricted. However, in providing error estimates it will

be very helpful.

On the sphere, we used the expansion (17.2) of in terms of spherical harmonics to

characterize positive de¬nite kernels. Spherical harmonics can also be interpreted as the

system of eigenfunctions of the Laplace“Beltrami operator on S d’1 . Hence, a possible way

of generalizing (17.2) would be to use the eigenfunctions of the Laplace“Beltrami operator

on M; these form an orthonormal basis of L 2 (M). But any other orthonormal basis of

L 2 (M), which is also dense in C(M), will do.

17.4 Interpolation on compact manifolds 319

Theorem 17.19 Let M be a compact C k Riemannian manifold and let {Y }∞ be an or-

=1

thonormal basis for L 2 (M), which is also dense in C(M). Suppose that

∞

(x, y) = a Y (x)Y (y)

=1

is in C(M — M). Then is positive semi-de¬nite on M if and only if all the coef¬cients a

are nonnegative. Moreover, if they are all positive then is positive de¬nite.

Proof The proof is more or less the same as the proof for Theorem 17.8. Hence, we leave

the details for the reader.

After having characterized positive de¬nite kernels on manifolds, we come to error

estimates. Of course, we could split the interpolation error again, using the power function;

then the next step would be to use the optimality of the power function with respect to its

coef¬cients. Hence, if we wanted to follow this path we would have to construct a better-

suited family of functions, which reproduce polynomials and have uniformly bounded

Lebesgue functions. In fact, though, we want to use a different approach based on Proposition

17.18. To this end we need to relate the distance measure on the manifold to that in the range

of the charts. As pointed out earlier, the main implication of this is that the dist topology is

equivalent to the initial topology.

Lemma 17.20 Let M be a C k Riemannian manifold with k ≥ 1. For every x ∈ M there

exists a chart (Ux , •x ) with x ∈ Ux and constants 0 < m x ¤ Mx such that

m x •x (y) ’ •x (z) ¤ dist(y, z) ¤ Mx •x (y) ’ •x (z) 2 , y, z ∈ Ux .

2

’1

Moreover, •(Ux ) can be chosen as {v ∈ Rn : v < r } with r = r x > 0. Finally, •x is in

2

C k up to the boundary of •x (Ux ).

Proof Let (U, •) be a chart with x ∈ U . Without restriction we can assume that •(x) = 0.

Moreover, there exists an r > 0 such that the closed ball B(0, 3r ) = {v ∈ Rn : v 2 ¤ 3r }

is contained in V := •(U ). We de¬ne V := {v ∈ Rn : v 2 < r } and U := • ’1 (V ).

Next, we note that the function

‚• ’1 ‚• ’1

n

(y, ±) ’ ± j ±k g y (•(y)), (•(y))

‚v j ‚vk

j,k=1

is continuous on the compact set • ’1 (B(0, 3r )) — S n’1 and hence attains a minimum and a

maximum. Moreover, since g y is positive de¬nite, the minimum is positive. In other words,

there exist constants 0 < m x ¤ Mx < ∞ such that

‚• ’1 ‚• ’1