Mathematics – Differential Geometry
Scientific paper
2012-02-17
Mathematics
Differential Geometry
24 pages, 2 figures
Scientific paper
Let $f:M^{2m}\to \R^{2m+1}$ be a hypersurface which allow only generic corank one singularities, and admits globally defined unit normal vector field $\nu$. Then we show the existence of an index formula. For example, when $m=1$ or $m=2$, it holds that 2 deg(\nu)=\chi(M^2_+)-\chi(M^2_-)+\chi(A^+_3)-\chi(A^-_3), 2 deg(\nu)=\chi(M^4_+)-\chi(M^4_{-})+\chi(A^+_3)-\chi(A^-_3)+\chi(A^+_5)-\chi(A^-_5), respectively, where $M^{2m}_+$ (resp. $M^{2m}_-$) is the subset of $M^{2m}$ at which the co-orientation of $M^{2m}$ induced by $\nu$ coincides (resp. does not coincide) with the orientation of $M^{2m}$ for $m=1$, 2, and $\chi(A^+_{2k+1})$ (resp. $\chi(A^-_{2k+1})$) for $k=1,2$ is the Euler number of the set of positive (resp. negative) $\A_{2k+1}$-singular points of $f$. The formula for $m=1$ is known. To prove the results, we prepare an index formula for corank one singularities of vector bundle homomorphisms on $M^{2m}$. As its application, an index formula for Blaschke normal map of strictly convex hypersurfaces in $R^{2m+1}$ is also given.
Saji Kentaro
Umehara Masaaki
Yamada Kotaro
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