By James R. Munkres

A readable advent to the topic of calculus on arbitrary surfaces or manifolds. available to readers with wisdom of uncomplicated calculus and linear algebra. Sections comprise sequence of difficulties to enhance concepts.

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2. B1 -surfaces: F (x) = const, G,y (y) = 0. Without loss of generality we can put locally F = 0 and G = y which implies K = −1/y 2 . 3. B2 -surfaces: F,x (x) = 0, G,y (y) = 0. Without loss of generality we can put locally F = x and G = y, which implies K = −1/(x + y)2 . 64) reads (E + E¯−1 ) E,xy + E,x + E,y 2(x + y) = 2E,x E,y . 72) This equation is obviously closely related to the Ernst equation. Both equations can be obtained as reductions of the following system for two complex potentials E1 and E2 , E1,x + E1,y 2(x + y) E2,x + E2,y (E1 + E2 ) E2,xy + 2(x + y) (E1 + E2 ) E1,xy + = 2E1,x E1,y , = 2E2,x E2,y .

A transformation of the form φ = φ + ωt with constant ω describes a change to a coordinate system rotating with ω with respect to the original system. The corresponding transformation of the metric coeﬃcients is gtt = gtt + 2ωgtφ + ω 2 gφφ , gtφ = gtφ + ωgφφ . To require that g should be Minkowskian on the symmetry axis does not ﬁx ω. 24 2 The Ernst Equation In the following we are particularly interested in the study of the gravitational ﬁeld of isolated matter conﬁgurations in an otherwise empty universe.

E. for β = 0. In this case we get with the above formulas E = E¯ = e2U with U =− 1 4πi ln α Γ1 (K − ζ)2 + 2 dK . 23) Thus, all solutions are real in this case which implies that they belong to the static Weyl class. 23) solves the axisymmetric Laplace equation. 1) that the contour integral there is equivalent to the Poisson integral with a distributional density. 23) that the dependence on the physical coordinates and ζ is exclusively via the branch points of the family of surfaces L0 . 6), it is possible to reduce the above matrix Riemann–Hilbert problem further.