Banach-Stone type theorems for C¹-function spaces over Riemannian manifolds

Abstract

Let M be a compact Riemannian manifold and let C¹(M, ℝ^d) be the space of all C¹-maps of M to the d-dimensional Euclidean space ℝ^d with the C¹-topology. For p ∈ [1, ∞] and a compact submanifold K of M, we define a norm

$${\parallel \cdot \parallel_{\left\langle {M,K;p} \right\rangle }}\,on\,{C^1}\left( {M,{^d}} \right)\,by\,{\parallel f \parallel_{\left\langle {M,K;p} \right\rangle }} = {\left( {\parallel {fK} \parallel_\infty ^p + \parallel {Df} \parallel_\infty ^p} \right)^{1/p}}$$

for f ∈ C¹(M, ℝ^d), where Df denotes the derivative off (the norm Df∞ will be defined in Section 1). For two pairs (M, K), (N, L) of Riemannian manifolds and their submanifolds, we characterize a surjective linear •_<M,K;pî - •_<N,L;p>-isometry T: C¹(M, ℝ^d) → C¹(N,ℝ^d), satisfying some regularity conditions, as a modified weighted composition operator whose symbol is a Riemannian homothety ϕ: N → M. If K and L are not singletons, then ϕ is a Riemannian isometry and satisfies ϕ(L) = K. The result indicates that the isometries of C¹-function spaces with respect to the above norms determine not only the isometry type of the ambient manifold but also the embedding type of the submanifolds up to isometry. Applying this result we study deformations of isometry groups associated with some perturbations of norms on C¹(M,∝^d). Aspects of these deformations naturally depend on isometry groups of the underlying manifolds.