As we know, the square $S=[0,1]\times[0,1]$ is not a manifold with boundary. Instead, it's a manifold with corners. For a tangent vector field on a compact manifold with boundary, we have the Poincaré–Hopf index theorem.

Then here is my question: is there a similar index theorem for manifolds with corners (e.g., the square, the cube) which relates the indices of a tangent vector field to the Euler characteristics of the manifold? If affirmative, what is the statement of this theorem?

Btw, I find an economics textbook (Vives, X., 1999. Oligopoly pricing: old ideas and new tools. MIT Press) which states that the Poincaré–Hopf theorem can apply to a compact cube (a manifold with corners). The statement (on page 362) is as below:

This "version" is frequently used in economics papers. Is it correct?

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