This paper deals with the basic notions of k-tautimmersions . These notions come
from two special cases; that is, tight and taut immersions. Tight and taut based on high and distance functions respectively and their basic notions are normal bundle, endpoint map, focal point, critical normal. We generalize hight and distance functions to cylindrical function and define basic notions of k-taut immersions such as k-plane
normal bundle , end k-plane map, focal k-plane, and critical k-plane normal. Then we
prove index theorems for cylindrical function similar to the standard index theorems
of distance function . In this way, the key point is the relation between focal point and focal k-plane