  Multivariate functional data has received considerable attention but testing
for equality of mean surfaces and its profile has limited progress. The
existing literature has tested equality of either mean curves of univariate
functional samples directly, or mean surfaces of bivariate functional data
samples but turn into functional curves comparison again. In this paper, we aim
to develop both the profile and globe tests of mean surfaces for two-sample
bivariate functional data. We present valid approaches of tests by employing
the idea of pooled projection and by developing a novel profile functional
principal component analysis tool. The proposed methodology enjoys the merit of
readily interpretability and implementation. Under mild conditions, we derive
the asymptotic behaviors of test statistics under null and alternative
hypotheses. Simulations show that the proposed tests have a good control of the
type I error by the size and can detect difference in mean surfaces and its
profile effectively in terms of power in finite samples. Finally, we apply the
testing procedures to two real data sets associated with the precipitation
change affected jointly by time and locations in the Midwest of USA, and the
trends in human mortality from European period life tables.
