A Comprehensive Introduction to Differential Geometry, Vol. by Michael Spivak

By Michael Spivak

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Additional resources for A Comprehensive Introduction to Differential Geometry, Vol. 4, 3rd Edition

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96)). are functions. Using this explicit formula obtained produced with MathernaticaTM. 6 are 3. Bonnet Surfaces 56 Fig. 5. 6. Another Bonnet surface of Fig. 8 show presents a (J = 0). Figure Both have the Fig. 7. 8 presents same a Bonnet surfaces of type H(O) = an umbilic 0, Ho = Maximal Bonnet surface H(s) solution surface with initial data critical point with J was = 0 used. Our numerical H(O) E R arbitrary, Ho H(s) according to (3-120). Take numbers a images of surface with non-umbilic critical point of the Fig.

25) of Bonnet a Then, of the following five forms: hi (w) = h4 (W) = = -ie 4i 1, tanh(2w), h5(W) = tan(2w). W right = V(W) h' (w) hl(w) hand side is a scaling transformation to normalize h(w). 26) h'(w)). 12), (3-25), h2(W) W, by and up to the normalizations Proof. 24) aER, bEC. 4. Let F be of R,,. 20) are invariant with respect to family FT -+ aYT with a simultaneous change of the w-+aw+b and thus 0 , check the surface conformal coordinate We will not -+ ) (h(w) + h(w)) = 2 (h'(w) - obviously harmonic.

Ranscendents Ignore for the matrices (3-70). 6). 2) one obtains that the Hazzidakis equation for Bonnet B surfaces is equivalent to a certain Painlev6 VI equation. ) Equation (3. 4. 1. ( H/(X) 2 x 2 2 + x - 1 ) possesses the H2 (X) Xhi (X) + first integral + 2- + 2 (x - 1)2 7JI(X) H(X) (X + 1) 2 (x 1) 02. 73). 75) which Y(X) the 1 (X 12 y - y any solution of (Pvj) 1 + root - solves the Painlev6 VI equation d y fixed a 1) W"(x) + (0 x (0 2))H'(x) 71 W + (x 1) IV W (x (x 2 Y(X) 0. 74) and id. 73).

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