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Given a surface defined as the graph of a real polynomial in two variables, we analyze some basic subsets characterized by its tangential singularities. If the parabolic curve is compact we provide certain criteria to determine when the unbounded component of its complement is hyperbolic. Moreover, we obtain an upper bound of the number of Gaussian cusps that holds even if the parabolic curve is non compact. [ABSTRACT FROM AUTHOR]

Loewner's umbilic-index conjecture cannot be extended to surfaces immersed in ℝ4. More precisely, given n ∈ ℤ, there exists an analytic surface immersed in ℝ4 having an isolated umbilic point of index n/2. [ABSTRACT FROM AUTHOR]