ACTA issues

Banach--Stone type theorems for $C^1$-function spaces over Riemannian manifolds

Kazuhiro Kawamura

Acta Sci. Math. (Szeged) 83:3-4(2017), 551-591

Abstract. Let $M$ be a compact Riemannian manifold and let $C^{1}(M,\mathbb{R}^{d})$ be the space of all $C^1$-maps of $M$ to the $d$-dimensional Euclidean space $\mathbb{R}^d$ with the $C^1$-topology. For $p \in[1,\infty ]$ and a compact submanifold $K$ of $M$, we define a norm $\| \cdot\| _{\langle M,K;p\rangle }$ on $C^{1}(M,\mathbb{R}^{d})$ by $ \| f \| _{\langle M,K;p\rangle } = (\| f|K \| _{\infty }^{p} + \| Df \| _{\infty }^{p})^{1/p} $ for $f \in C^{1}(M,\mathbb{R}^{d})$, where $Df$ denotes the derivative of $f$ (the norm $\| Df \| _{\infty }$ will be defined in Section 1). For two pairs $(M,K)$, $(N,L)$ of Riemannian manifolds and their submanifolds, we characterize a surjective linear $\| \cdot\| _{\langle M,K;p\rangle }-\| \cdot\| _{\langle N,L;p\rangle }$-isometry $T\colon C^{1}(M,\mathbb{R}^{d}) \to C^{1}(N,\mathbb{R}^{d})$, satisfying some regularity conditions, as a modified weighted composition operator whose symbol is a Riemannian homothety $\varphi\colon N \to M$. If $K$ and $L$ are not singletons, then $\varphi $ is a Riemannian isometry and satisfies $\varphi(L) = K$. The result indicates that the isometries of $C^1$-function spaces with respect to the above norms determine not only the isometry type of the ambient manifold but also the {\it embedding type} of the submanifolds up to isometry. Applying this result we study deformations of isometry groups associated with some perturbations of norms on $C^{1}(M,\mathbb{R}^{d})$. Aspects of these deformations naturally depend on isometry groups of the underlying manifolds.

DOI: 10.14232/actasm-016-323-4

AMS Subject Classification (1991): 46E15, 55R25, 53C99

Keyword(s): isometry, weighted composition operator, Banach--Stone theorem, Banach bundle, continuously differentiable functions, Riemannian manifold, embedding

Received December 25, 2016, and in revised form April 24, 2017. (Registered under 73/2016.)