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By Luther Pfahler Eisenhart

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8. Vector Fields Along Curves Without further qualification, a curve in a manifold M always means for us a smooth, parametrized curve; that is, a smooth map γ : I → M , where I ⊂ R is some interval. Unless otherwise specified, we won’t worry about whether the interval is open or closed, bounded or unbounded. A curve segment is a curve whose domain is a closed, bounded interval [a, b] ⊂ R. If γ : I → M is a curve and the interval I has an endpoint, smoothness of γ means by definition that γ extends to a smooth curve defined on some open interval containing I.

In coordinates, ω # has components ω i := g ij ωj , where, by definition, g ij are the components of the inverse matrix (gij )−1 . One says ω # is obtained by raising an index. Probably the most important application of the sharp operator is to extend the classical gradient operator to Riemannian manifolds. If f is a smooth, real-valued function on a Riemannian manifold (M, g), the gradient of f is the vector field grad f := df # obtained from df by raising an index. Looking through the definitions, we see that grad f is characterized by the fact that df (Y ) = grad f, Y for all Y ∈ T M , and has the coordinate expression grad f = g ij ∂i f ∂j .

N−1 , −τ, ξ n ), taking the hemisphere {τ < 0} to the hemisphere {ξ n > 0}. This shows that κ is a conformal map, and therefore it suffices to show that h3R (κ∗ V, κ∗ V ) = h2R (V, V ) for a single strategically chosen vector V at each point. Do this for V = ∂/∂v. 12) of κ: Compute the pullback in complex notation, by noting that h3R = R2 dz d¯ z , (Im z)2 h2R = 4R4 dw dw ¯ , (R2 − |w|2 )2 and using the fact that a holomorphic diffeomorphism z = F (w) is a conformal map with F ∗ (dz d¯ z ) = |F (w)|2 dw dw.

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