An explicit proof of the generalized Gauss-Bonnet formula
From probability to geometry (II) - Volume in honor of the 60th birthday of Jean-Michel Bismut, Astérisque, no. 328 (2009), pp. 137-160 
@incollection{AST_2009__328__137_0,
     author = {Gillet, Henri and \"Unl\"u, Fatih M.},
     title = {An explicit proof of the generalized {Gauss-Bonnet} formula},
     booktitle = {From probability to geometry (II) - Volume in honor of the 60th birthday of Jean-Michel Bismut},
     editor = {Dai Xianzhe and L\'eandre R\'emi and Xiaonan Ma and Zhang Weiping},
     series = {Ast\'erisque},
     pages = {137--160},
     year = {2009},
     publisher = {Soci\'et\'e math\'ematique de France},
     number = {328},
     mrnumber = {2664468},
     zbl = {1235.32006},
     language = {en},
     url = {https://www.numdam.org/item/AST_2009__328__137_0/}
}
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Gillet, Henri; Ünlü, Fatih M. An explicit proof of the generalized Gauss-Bonnet formula, dans From probability to geometry (II) - Volume in honor of the 60th birthday of Jean-Michel Bismut, Astérisque, no. 328 (2009), pp. 137-160. https://www.numdam.org/item/AST_2009__328__137_0/

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