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Let [tex] r(u_i)[/tex] be a surface with i=1,2. Suppose that its first fundamental form is given as

[tex] ds^2 = a^2(du_1)^2 + b^2(du_2)^2[/tex]

which means that if [tex] r_1 = ∂r/∂u_1[/tex] and [tex] r_2= ∂r/∂u_2[/tex] are the tangent vectors they satisfy

[tex] r_1.r_2 = 0[/tex]

[tex] r_1.r_1 = a^2[/tex]

[tex] r_2.r_2 = b^2[/tex]

My aim is to is to calculate the second fundamental form (whose answer I know but is a boring expression so I won't write here). The most direct approach is to calculate the levi-civita connection coefficients from the metric and then calculate curvature and then scalar curvature which will be Gauss curvature upto a factor (since it is a surface). But this is very oafy calculation.

The other way is to somehow calculate the second fundamental form whose coefficients are given by [tex] (r_{ij},N)[/tex] where [tex] r_{ij} = ∂^2r/∂u_i∂u_j[/tex] and N is normal to the surface (generically given by) [tex] r_1 \times r_2[/tex] . Then using this and the metric one can calculate Gauss curvature (as the eigenvalues of G^{-1}B where G is the first and B is the second fundamental form).

As a motivation I tried a particular simple example where the normal vector is r/|r| itself (for instance sphere). Then taking derivatives of the equation 0 = r_{1}.r and using equalities (that come from the assumption of the metric) [tex] a.a_i = r_1.r_{1i}, b.b_i = r_2.r_{2i}[/tex] it is a matter of seconds to find the second fundamental form (in fact off diagonal terms vanish etc). However normal vector is not necessarily r itself in the case of general diagonal metric. All we can say is [tex] r_1.r_2 = 0[/tex] which again by derivation yields some equations but does not help me because I don't know any other way to express the normal vector other than r_1 x r_2 which is useless to do component calculations (if there was a way to turn products of the form [tex] (r_1 \times r_2).r_{ij}[/tex] into serioes of dot products that would also be enough but as far as I know there is no such formula and you can only exchange the places of factors). So my question is can you think of anyway to express N in some particular simple way in such a metric. I am really hoping that there should be a way to do this simply.

Thanks

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# Second fundamental form of surface with diagonal metric

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