# Asymptotic Behavior of Mass and Spacetime Geometry by F. J. Flaherty

By F. J. Flaherty

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1. For any V ∈ TP M , the directional derivative DV n(P ) ∈ TP M . , for any U, V ∈ TP M , we have (∗) SP (U) · V = U · SP (V) SP is called the shape operator at P . Proof. For any curve α : (−ε, ε) → M with α(0) = P and α′ (0) = V, we observe that n◦ α has constant length 1. 1 of Chapter 1, DV n(P ) · n(P ) = (n◦ α)′ (0) · (n◦ α)(0) = 0, so DV n(P ) is in the tangent plane to M at P . 3 of the Appendix. 46 Chapter 2. Surfaces: Local Theory Symmetry is our first important application of the equality of mixed partial derivatives.

1 suggests that we define the second fundamental form, as follows. If U, V ∈ TP M , we set IIP (U, V) = SP (U) · V. Note that the formula (†) on p. 45 shows that the curvature of the normal slice in direction V (with V = 1) is, in our new notation, given by ±κ = −DV n(P ) · V = SP (V) · V = IIP (V, V). As we did at the end of the previous section, we wish to give a matrix representation when we’re working with a parametrized surface. 1, we have ℓ = IIP (xu , xu )= −Dxu n · xu = xuu · n m = IIP (xu , xv )= −Dxu n · xv = xvu · n = xuv · n = IIP (xv , xu ) n = IIP (xv , xv ) = −Dxv n · xv = xvv · n.

6 will establish (see Exercise 13) that there is a continuous function θ˜: ∆ → R so that h(s, t) = ˜ t), sin θ(s, ˜ t) for all (s, t) ∈ ∆. 6 that cos θ(s, C ˜ L) − θ(0, ˜ 0) = θ(0, ˜ L) − θ(0, ˜ 0) + θ(L, ˜ L) . 4. 4 through which the position vector of the curve turns, starting at 0 and ending at π; since the curve lies in the upper half-plane, we must have N1 = π. But N2 is likewise the angle through which the negative of the position vector turns, so N2 = N1 = π. With these assumptions, we see that the rotation index of the curve is 1.