The Taylor Expansion of the Exponential Map and Geometric Applications
  In this work we consider the Taylor expansion of the exponential map of a submanifold immersed in R^n up to order three, in order to introduce the concepts of lateral and frontal deviation. We compute the directions of extreme lateral and frontal deviation for surfaces in R^3. Also we compute, by using the Taylor expansion, the directions of high contact with hyperspheres of a surface immersed in R^4 and the asymptotic directions of a surface immersed in R^5. 
M. G. Monera, A. Montesinos-Amilibia and E. Sanabria-Codesal
math.DG