Unformatted text preview: ◦ He called it the total curvature ; we call it the Gaussian curvature . • Riemann ◦ He was a student of Gauss. His work extended Gauss’s and was first delivered in his Habilitationsvortrag . ◦ He was interested in the empirical question of exactly what kind of space we live in. How do we know its flat? ◦ Introduced study of m-dimensional manifolds immersed in n-dimensional space. ◦ Focused on the intrinsic nature of the manifold, not on specific coordinate system. ◦ Riemannian metric ⋆ Defined on an n-dimensional manifold parametrized by coordinates x 1 ,x 2 ,...,x n . ⋆ An n × n matrix G = (g ij ) whose entries are functions of x 1 ,x 2 ,...,x n . This matrix must be positive definite , which means that if you multiply this matrix on the left by a nonzero row vector and on the right by the corresponding column vector the result is positive. ⋆ The square of the length of an “infinitesimal” segment from (x 1 ,x 2 ,...,x n ) to (x 1 + dx 1 ,x 2 + dx 2 ,...,x n...
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This note was uploaded on 12/29/2011 for the course MATH 378 taught by Professor Wen during the Fall '10 term at SUNY Stony Brook.
- Fall '10