  The usual interpretation of Weyl geometry is modified in two senses. First,
both the additive Weyl connection and its variation are treated as (1, 2)
tensors under the action of Weyl covariant derivative. Second, a modified
covariant derivative operator is introduced which still preserves the tensor
structure of the theory. With its help, the Riemann tensor in Weyl geometry can
be written in a more compact form. We justify this modification in detail from
several aspects and obtain some insights along the way. By introducing some new
transformation rules for the variation of tensors under the action of Weyl
covariant derivative, we find a Weyl version of Palatini identity for Riemann
tensor. To derive the energy-momentum tensor and equations of motion for
gravity in Weyl geometry, one naturally applies this identity at first, and
then converts the variation of additive Weyl connection to those of metric
tensor and Weyl gauge field. We also discuss possible connections to the
current literature on Weyl-invariant extension of massive gravity and the
variational principles in f(R) gravity.
