Good morning/day/evening/night,
I was presented to the generalized Gauss-Bonnet-Chern theorem for hypersurfaces in Euclidean space;
For a closed, even dimensional manifold $M$ with dimension $n$ embedded in $\mathbb{R}^{n+1}$ we have $\int_M K\mathbb dV = \text{Volume }\mathbb{S}^n\cdot\frac{\chi(M)}{2}.$
I wonder,
- How does this theorem look for compact manifolds with boundary?
- Can this theorem be generalized to hold for closed odd dimensional manifolds? (I assume there will be a problem here since the Euler characteristic vanishes.)
Both questions are for hypersurfaces in Euclidean space.
Any reference will be greatly appreciated.