Inflection points and topology of surfaces in 4-space

Garcia, Ronaldo; Mochida, Dirce; Romero Fuster, Maria; Soares Ruas, Maria

doi:10.1090/S0002-9947-00-02404-1

Inflection points and topology of surfaces in 4-space
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by Ronaldo Alves Garcia, Dirce Kiyomi Hayashida Mochida, Maria del Carmen Romero Fuster and Maria Aparecida Soares Ruas PDF

Trans. Amer. Math. Soc. 352 (2000), 3029-3043 Request permission

Abstract:

We consider asymptotic line fields on generic surfaces in 4-space and show that they are globally defined on locally convex surfaces, and their singularities are the inflection points of the surface. As a consequence of the generalized Poincaré-Hopf formula, we obtain some relations between the number of inflection points in a generic surface and its Euler number. In particular, it follows that any 2-sphere, generically embedded as a locally convex surface in 4-space, has at least 4 inflection points.

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