The Index Theorem from Gauss-Bonnet and Riemann-Roch Theorem

• Main Authors: Yu-Chan Chang, 張育展
• Other Authors: Hung-Lin Chiu
• Format: Others
• Language: en_US
• Published: 2010
• Online Access: http://ndltd.ncl.edu.tw/handle/92526436832807455591

碩士 === 國立中央大學 === 數學研究所 === 98 === In this thesis, we prove two important theorems in geometry. In chapter one, we state the Gauss-Bonnet theorem on even dimensional manifold and give the detail of the proof of two dimensional case. The proof is based on the paper "A simple intrinsic proof of the Gauss-Bonnet formula for closed Riemannian manifold", published by S.S. Chern in 1943. A little history of this theorem is included. Chapter two and three mainly focus on Riemann-Roch theorem on one-dimensional complex manifold, Riemann surface. We establish some basics on Riemann surface in chapter two, such as divisors, holomorphic line bundles, sheaves and cohomology on sheaves, also Hodge theorem in the end of this chapter. The proof of Riemann-Roch is in the chapter three. In chapter four, we show a theorem by calculating two analytic indices of two operators, which give us Gauss-Bonnet and Riemann-Roch theorem. This theorem is the Atiyah-Singer index theorem, proved by Atiyah and Singer in 1963.