Celebratio Mathematica — Chern

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Works connected to Edwin H. Spanier

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article S.-S. Chern and E. Spanier: “The homology structure of sphere bundles,” Proc. Natl. Acad. Sci. U.S.A. 36 : 4 (April 1950), pp. 248–255. MR 0035994 Zbl 0035.38803

article S.-S. Chern and E. Spanier: “A theorem on orientable surfaces in four-dimensional space,” Comment. Math. Helv. 25 (1951), pp. 205–209. MR 0044883 Zbl 0043.38403

Let \( M \) be a closed oriented surface differentiably imbedded in a Euclidean space \( E \) of four dimensions. Let \( G \) denote the Grassmann manifold of oriented planes through a fixed point \( O \) of \( E \). It is well known that \( G \) is homeomorphic to the topological product \( S_1\times S_2 \) of two 2-spheres. By mapping each point \( P \) of \( M \) into the oriented plane through \( O \) parallel to the oriented tangent plane to \( M \) at \( P \), we define a mapping \( t:M\to G \). If \( M \), \( S_1 \), \( S_2 \) denote also the fundamental cycles of the respective manifolds and \( t_* \) denotes the homomorphism induced by \( t \), we have \[ t_*(M) \sim u_1S_1 + u_2S_2. \] In a recent paper [1949] Blaschke studied the situation described above by methods of differential geometry and proved that the sum \( u_1 + u_2 \) equals the Euler charateristic of \( M \). He also asserted that \( u_1 = u_2 \). The object of this note is to give a proof of this assertion, as well as a new proof of the theorem on \( u_1+u_2 \).