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Please note that the content of this book primarily consists of articles available from Wikipedia or other free sources online. In mathematics, a Riemann Roch theorem for smooth manifolds is a version of results such as the Hirzebruch Riemann Roch theorem or Grothendieck Riemann Roch theorem (GRR) without a hypothesis making the smooth manifolds involved carry a complex structure. Results of this kind were obtained by Michael Atiyah and Friedrich Hirzebruch in 1959, reducing the requirements to something like a spin structure.