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\title{Geometry}
\author{Last Updated: November 1986}
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Riemannian metrics and Riemannian manifolds. Affine connections and
Riemannian (\textit{i.e.}, Levi-Civita) connections. Cristoffel
symbols, Geodesics and the exponential map. The Riemann curvature
tensor. Sectional curvature. Connection with Gaussian curvature of
surfaces. Ricci and scalar curvature. Jacobi vector fields and
conjugate points. Complete manifolds. Classification of complete
manifolds of constant curvature. The first and second variation of
the energy and applications. Basic properties of Lie groups and Lie
algebras: correspondence between Lie groups and Lie algebras,
subgroups and subalgebras, etc. Examples of Lie groups and their Lie
algebras. Affine connections and curvature for a Lie group with
bi-invariant metric.
In addition to learning definitions and theorems, the student will be
expected to apply the theorems in simple, concrete cases, such as, for
example, to show that the geodesics on a round $2$-sphere are great
circles.
\paragraph{Theorems:}
Existence of Riemannian metric and Riemannian connection. Gauss'
Lemma. The Hopf-Rinow Theorem. The Bonnet-Myer Theorem. The
Frobenius Theorem. Left-invariant and bi-invariant metrics and volume
elements on Lie groups.
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