e/Riemann–Roch theorem for smooth manifolds

Information
has gloss	eng: 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.
lexicalization	eng: Riemann-Roch theorem for smooth manifolds
lexicalization	eng: Riemann–Roch theorem for smooth manifolds
instance of	c/Algebraic surfaces