Question

P(u,v)=(f(v)cos(u),f(v)sin(v),g(v))

Find formulas for the Christoffel symbols, the second fundamental form, the shape operator, the Gaussian curvature and the mean curvature.

Answer 1

1. In differential geometry, the second fundamental form (or shape tensor) is a quadratic form on the tangent plane of a smooth surface in the three-dimensional Euclidean space, usually denoted by   (read "two"). Together with the first fundamental form, it serves to define extrinsic invariants of the surface, its principal curvatures. More generally, such a quadratic form is defined for a smooth hypersurface in a Riemannian manifold and a smooth choice of the unit normal vector at each point.

2. The Shape operator and the second fundamental form: Let M : X : U → R3 be a surface. The map n : M → S2 ⊂ R3, p → n(p) is called the Gauss map, where n(p) is the unit norm at p = X(u) where u = (u1,u2
). It is very important and easy to check that the image of dnp : TpM → Tn(p)R3 in fact lies in TpM, the tangent space of M. So dnp : TpM → TpM is a linear transformation. Moreover, Sp := −dnp is symmetric, i.e., for any u, v ∈ TP (M), we have SP (u) · v = u · SP (v).
The linear map Sp = −dnp : TpM → TpM is called the Shape operator.
The bilinear form (quadratic form) IIp on TpM × TpM → R defined by Sp(X) · Y is called the second fundamental form. Remark that from linear algebra, every symmetric bilinear form II on an inner product space V corresponds to a unique symmetric linear transformation T : V → V in such way that T(v) · w = II(v, w) for every v, w ∈ V . So the second fundamental form and the shape operator
determines each other in a conventional way.
In terms of the local coordinates, the matrix representation of IIp with respect to basis
{X1, X2} is hij = Sp(Xi) · Xj = −ni· Xj = n · Xij ,It is easy to see that the the matrix of the shape operator SP: TP (M) → TP (M)with respect to the basis {X1, X2} is A = (gij )−1(hij ).

3. The Gauss Curvature and Mean Curvature: K = κ1κ2, H = (κ1 + κ2)/2. We
also have
Alternatively, the Gauss curvature of surface M at the point p is defined by
K(p) = det(Sp).

Hence K(p) = det(hij )det(gij )−1

Remark: Compare some notations:
Modern: hij = Xij · n = SP (Xi) · Xj (classical: l = Xuu · n, m = Xuv · n, n = Xvv · n.)