A proof of the Chern-Lashof conjecture in dimensions greater than five

Abstract

Chern and Lashof ([1], [2]) conjectured that if a smooth manifoldM m has an immersion intoR w, then the best possible lower bound for its total absolute curvature is the Morse number μ(M). We give a proof of this whenm>5. Under the same dimension restriction, our methods allow us to show that μ(M) is still the best possible lower bound among immersions within a fixed regular homotopy class except in the casew=m+1=even, for which the best lower bound is max {μ(M), 2 |d|}, whered the degree of the Gauss map.