Abstract
In this paper, we discuss the equality between invariant quantities, such as the Teichmüller distance and the Carathéodory distance, in Teichmüller spaces. We shall give some conditions under which these quantities are the same at some points. Examples of Fuchsian groups will be constructed in relation to the conditions above. In particular, we shall construct a Teichmüller disk in T(1) determined by a quadratic differential with a simple zero where the Teichmüller distance and the Carathéodory distance are the same. We also give an estimate of the variation of critical exponent quasi-Fuchsian groups in terms of Grunsky coefficients.
