ACTA issues

Isometries of function spaces over Riemannian manifolds

Kazuhiro Kawamura

Acta Sci. Math. (Szeged) 83:3-4(2017), 593-617

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 $\mathbb{R}^d$. For $p \in[1,\infty ]$, we introduce a norm $\| \cdot\| _{D,p}$ on $C^{1}(M,\mathbb{R}^{d})$ of the form $ \| f \| _{D, p} = (\| f \| _{\infty }^{p} + \| Df \| _{\infty }^{p})^{1/p}, f \in C^{1}(M,\mathbb{R}^{d}) $, where $Df$ denotes the derivative of $f$ (see Section 1 for the precise definition of the norm). We prove that every surjective linear isometry $T\colon C^{1}(M,\mathbb{R}^{d})\to C^{1}(M,\mathbb{R}^{d})$ which maps the constant functions to the constant functions is a generalized weighted composition operator in that $ Tf(x) = U(f(\varphi(x))), f \in C^{1}(M,\mathbb{R}^{d}), x \in M $, for a Riemannian isometry $\varphi\colon M \to M$ of $M$ and a linear isometry $U\colon\mathbb {R}^{d} \to\mathbb {R}^{d}$. This is an analogue of the classical Banach--Stone theorem for $C^1$-function spaces and partly extends previous results [BotelhoJamison], [K3] to function spaces over higher dimensional spaces.

DOI: 10.14232/actasm-016-283-9

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

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

Received June 18, 2016, and in revised form March 19, 2017. (Registered under 33/2016.)