  Cook's [J. Roy. Statist. Soc. Ser. B 48 (1986) 133--169] local influence
approach based on normal curvature is an important diagnostic tool for
assessing local influence of minor perturbations to a statistical model.
However, no rigorous approach has been developed to address two fundamental
issues: the selection of an appropriate perturbation and the development of
influence measures for objective functions at a point with a nonzero first
derivative. The aim of this paper is to develop a differential--geometrical
framework of a perturbation model (called the perturbation manifold) and
utilize associated metric tensor and affine curvatures to resolve these issues.
We will show that the metric tensor of the perturbation manifold provides
important information about selecting an appropriate perturbation of a model.
Moreover, we will introduce new influence measures that are applicable to
objective functions at any point. Examples including linear regression models
and linear mixed models are examined to demonstrate the effectiveness of using
new influence measures for the identification of influential observations.
