Morse theory for analytic functions on surfaces Jaime Arango, Oscar Perdomo Abstract In this paper we deal with analytic functions f : S → R defined on a compact two dimensional Riemannian surface S whose critical points are semi degenerated (critical points having a non identically vanishing Hessian). To any element p of the set of semi degenerated critical points Q we assign an unique index which can take the values -1, 0 or 1, and prove that Q is made up of finitely many (critical) points with non zero index and embedded circles. Further, we generalize the famous Morse result by showing that the sum of the indexes of the critical points of f equals χ(S), the Euler characteristic of S. As an intermediate result we locally describe the level set of f near a point p ∈ Q. We show that the level set f -1 (f (p)) is either a) the set {p}, or b) the graph of a smooth curve passing through p, or c) the graphs of two smooth curves tangent at p or d) the graphs of two smooth curves building at p a cusp shape. Mathematics Subject Classification (2000): Primary 37B30 , Secondary 58E05, 30F30. Key words: Morse theory, analytic functions, Euler characteristics. 1 Introduction Let f : S → R be a smooth function on a compact oriented Riemannian surface. We say that p ∈ S is a critical point if df p : T p S → R vanishes for every v ∈ T p S. A critical point p ∈ S is non degenerated if the Hessian of f at p, hess(f )(p): T p S × T p S → R, is a non degenerated bilinear form, i.e. for every non zero vector v ∈ T p S, there exists a vector w ∈ T p S such that hess(f )(p)(v,w) = 0. A function f is called a Morse function if all its critical points are non degenerated. One of the most important achievements in geometry is Morse Theory, which, in the setting we deal with, establishes a relation between the nature of the critical points of a Morse function f : S → R and the topology of S, we have: Theorem 1 (Morse). Let f : S → R be a differentiable function on a compact ori- ented two dimensional surface S such that all its critical points are non degenerate. 1