Potentials for tangent tensor fields on spheroids

Abstract

In three-dimensional Euclidean space let S be a closed simply connected, smooth surface (spheroid). Let $\hat n$ be the outward unit normal to S, ▽ _S the surface gradient on S, I _S the metric tensor on S, g_ij the four covariant components of I _S (i,j = 1, 2), h _ij the four covariant components of -$\hat n$xI _S , and D _i covariant differentiation on S. It is well known that for any tangent vector field u on S there exist scalars ϕ and ψ on S, unique to within additive constants, such that $u = \nabla _s \varphi - \hat n \times \nabla _s \psi$; the covariant components of u are $u_i = D_i \varphi + h_i^j D_j \psi$. This theorem is very useful in the study of vector fields in spherical coordinates. The present paper gives an analogous theorem for real second-order tangent tensor fields F on S: for any such F there exist scalar fields H, L, M, N such that the covariant components of F are

$$F_{ij} = H h{}_{ij} + Lg_{ij} + E_{ij} (M,N),$$

where

$$E_{ij} (M,N) = ( - \nabla _s ^2 M)g_{ij} + 2D_i D_j M + (h_i ^k D_j + h_j ^k D_i )D_k N$$

It is shown that H and L are uniquely determined by F but that M and N are not. The set of complex scalar fields ℳ′ = M′+iN′ such that E _ij (M′, N′)=0 is shown to constitute a four-dimensional complex linear space $\mathfrak{W}$. The scalars M and N which help to generate a given F are uniquely determined by F and the condition that, for every ℳ′ in $\mathfrak{W}$,

$$\mathop \smallint \limits_s (M - iN)\mathcal{M}\prime dS = 0$$

The real linear space of second-order tangent tensor fields on S which have simultaneously the form E(M, 0) and the form E(0, N) is shown to have dimension zero on a sphere, dimension four on a non-spherical, intrinsically axisymmetric spheroid (a spheroid whose isometries form a compact, one parameter group), and dimension six on a spheroid which is not intrinsically axisymmetric. Applications of the representation theorem to tensor problems in spherical coordinates are briefly discussed.