Basic question about certain second order PDE's

[lavinia]

Given is the second order equation,

##X_{uv} = A(u,v)X_{u} + B(u,v)X_{v}## defined on a domain ##(u,v)## in the plane.

##X## is a three dimensional vector and ##A## and ##B## are arbitrary smooth functions.

When does such an equation determine a surface in ##R^3## and what in general can be said about the set of solutions?

Same question for ##X_{uu} = A(u,v)X_{u} + B(u,v)X_{v}##

 

[WWGD]

A standard result, the regular value theorem, is that if (u,v) is a regular value of a smooth function, then : https://en.wikipedia.org/wiki/Preimage_theorem , i.e., given a smooth map, the inverse image of a regular value is a submanifold. Let me see if I can apply that to your situation.

I assume you are using X=Y =##\mathbb R^3 ##

 

[lavinia]

A standard result, the regular value theorem, is that if (u,v) is a regular value of a smooth function, then : https://en.wikipedia.org/wiki/Preimage_theorem , i.e., given a smooth map, the inverse image of a regular value is a submanifold. Let me see if I can apply that to your situation.

I assume you are using X=Y =##\mathbb R^3 ##

Right. So the Jacobian of the map ##(u,v) \rightarrow X(u,v)## has to have rank 2 everywhere. Since each of the components of ##X(u,v)## is a scalar solution to the equation, this says something about the independence of these solutions. Do such triples of scalar solutions always exist?

Here is where this question came from. If one is given a parameterized surface, ##X(u,v)## in ##R^3## then one has the tangent vectors, ##X_{u}## and ##X_{v}## and the induced Riemannian metric, which in classical notation is ##X_{u}⋅X_{u} = E##, ##X_{u}⋅X_{v} = F##, ##X_{v}⋅X_{v} = G##.

##X_{u}## and ##X_{v}## are a basis for the tangent space and together with a positively oriented unit normal,##N##, form a moving frame. In terms of this frame one can write down the second derivatives of ##X(u,v)## and one gets three equations,

## X_{uu} = A_{11}(u,v)X_{u} + B_{11}(u,v)X_{v} + eN##
##X_{uv} = A_{12}(u,v)X_{u} + B_{12}(u,v)X_{v} + fN##
##X_{vv} = A_{22}(u,v)X_{u} + B_{22}(u,v)X_{v} + gN##

The ##A##'s and the ##B##'s are actually the Christoffel symbols of the Riemannian metric and ##e## ##f## and ##g## are the coefficients of the second fundamental form in ##(u,v)## coordinates. If ##f## is zero then the parameter curves ##u=## a constant and ##v=## are said to be conjugate.

But if ##f = 0## the one has the first differential equation that I asked about. So the second part of the question can be thought of as how much of a surface can be retrieved from the Christoffel symbols, ##A_{12}## and ##B_{12}## if the coordinate lines are assumed to be conjugate.

I suppose come to think of it, the answer is trivial. It is all surfaces expressed in conjugate coordinates. This was mentioned in Struik's book but I just didn't get it until reasoning through in this post.

BTW: ##A_{12} = Γ^{1}_{12}## ##B_{12} = Γ^{2}_{12}## ##A_{11} = Γ^{1}_{11}## ##B_{11} = Γ^{2}_{11}##