We define and study the secondary Chern-Euler class for a general submanifold of a Riemannian manifold. Using this class, we define and study the index for a vector field with non-isolated singularities on a submanifold. As an application, we give conceptual proofs of a result of Chern.

Keywords:

secondary Chern-Euler class, normal sphere bundle, Euler characteristic, index, non-isolated singularities, blow-up secondary Chern-Euler class, normal sphere bundle, Euler characteristic, index, non-isolated singularities, blow-up

MSC Classifications:

57R20 show english descriptions Characteristic classes and numbers 57R20 - Characteristic classes and numbers

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