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Tangential and Normal Euler Numbers,
Complex Points, and Singularities of Projections for Oriented
Surfaces in Four-Space

by Thomas F. Banchoff and Frank Farris

This article gives a geometric proof of the following.
Given a compact two-dimensional surface immersed smoothly in **R**^4
(or **C**^2), the sum of the Euler number of M, the Euler number of
its normal bundle, and the algebraic number of isolated *complex
points* is zero. Although this theorem has been examined and
proven in previous papers, the techniques required do not
illuminate the underlying geometry.