EQUATIONS OF ELASTIC HYPERSURFACES

16 SHELLS
If u = ψ(v) is the solution of (2.54) and v = ϕ(u) = ψ −1 (u) is the inverse function, we get
the system
u ′ = e cu a(ϕ(u)) , v ′ = e cu b(ψ(v))
on the line interval (α, β) ⊂ R, which should be integrated to get final answers.
Remark 2.11 The equality (2.47), which holds for suficiently small t, s ∈ (−ε, ε), shows
that the flow FU t (p) is a local algebra.
Moreover, the hypothesis (2.48) implies that the linear span of the vector fields U, V ∈
V (Ω) (over the field R) is a two-dimensional solvable Lie algebra. Sophus Lie developed
a theory of Lie groups to find whether constructions of solvable Lie algebra can contribute
into explicit integration of vector fields.
2.3 HYPERSURFACES
To formulate the basic definitions of hypersurface we need the following auxiliary definition.
Definition 2.12 Let m, n, k ∈ N be positive natural numbers, Ω be an open subset of R n and
F : Ω → R m be a C k -mapping. A point p ∈ U is said to be critical for F if the derivative
DF p : R n → R m is not of full rank, and regular if it is not critical.
A point q ∈ F (U) is said to be a regular value of F if every point of the pre-image
of {q} is regular and a critical value otherwise.
F −1 (q) := { p ∈ Ω : F (p) = q } (2.55)
Proposition 2.13 Let m, n ∈ N, Ω ⊂ R m and F : Ω → R n be a C k -mapping, k ≥ 1. The
following is equivalent:
i. p ∈ Ω is a regular point of F = (F 1 , · · · , F n ) : Ω → R n ;
ii.a. the gradients ∇F 1 , . . . , ∇F m of the coordinate functions F 1 , . . . , F n : Ω → R are
linearly independent at p, provided m ≥ n;
ii.b. the partial derivatives ∂ 1 F, . . . , ∂ m F of the vector-function F : Ω → R n are linearly
independent at p, provided n ≥ m;
iii.a. the derivative DF (p) := matr [ ∂ 1 F (p), . . . , ∂ m F (p) ] satisfies the condition
det [DF (p) · (DF ) ⊤ (p)] ≠ 0 at p, provided m ≥ n.
iii.b. the derivative DF (p) satisfies the condition det [ (DF ) ⊤ (p) · DF (p) ] ≠ 0 at p, provided
n ≥ m.
Proof: We propose the proof as an exercise.
Next definition of a hypersurface is basic in the present chapter and we formulate two
further definitions as well. The alternative definitions are very useful treating various problems
and later, in Lemma 2.19, we prove equivalence of all three definitions.
The next definition is most universal and can be used for manifolds.