Chapter 1 DIFFERENTIAL GEOMETRY OF REAL MANIFOLDS

1.2. RIEMANNIAN GEOMETRY 235
1. The subsets defined by Γγ : ψ2 = γ2, · · · , ψm = γm are the orthogonal trajectories to
the submanifolds Mc;
2. Γγ’s are geodesics after reparametrization by arc-length;
3. dψ ∧ dψ2 ∧ · · · ∧ dψm = 0, i.e., ψ, ψ2, · · · , ψm is a coordinate system.
Let Ψ = ( ∂ψ
∂x1
, · · · , ∂ψ
∂xm ) where x1, · · · , xm are coordinate functions on U. Let g = (gij) be
the matrix representation of the metric ds 2 relative to the coordinates x1, · · · , xm. We show
Lemma 1.2.3 With the above notation, a necessary and sufficient condition for the existence
of functions ψ2, · · · , ψm satisfying conditions 1, 2 and 3 is
• Ψg −1 Ψ ′ is expressible as a function of ψ only 4 ,
where superscript ′ denotes transpose.
Proof - First we show the necessity. Let h = (hij) denote the the transformed metric relative
to the coordinate system ψ, ψ2, · · · , ψm. Since the orthogonal trajectories ψ2 = γ2, · · · , ψm =
γm are geodesics (after reparametrization) the metric h = (hij) has the property
1. h11 is a function of ψ only;
2. h1i = hi1 = 0.
It follows from the transformation property of the metric that the 11-entry of the symmetric
matrix h = A −1 gA ′−1 is expressible as function of only ψ and its 1i entries vanish for i > 1.
Here A denotes the matrix
⎛
⎜
A = ⎜
⎝
∂ψ
∂x1
∂ψ
∂x2
.
∂ψ
∂xm
∂ψ2
∂x1
∂ψ2
∂x2
.
∂ψ2
∂xm
Taking inverse of A −1 gA ′−1 we obtain the necessity. To prove sufficiency let ψ2, · · · , ψm be
a coordinate system on Mc (c fixed). We want to transport this coordinate system to Mb for
4 In other words, ||gradψ|| is constant on each hypersurface ψ = c in which case Γγ’s are the trajectories
of the vector field gradψ. This formulation of the geodesic equations is the Hamilton-Jacobi view point
which was introduced in the subsection on symplectic and contact structures in chapter 1. We prove the
validity of this formulation directly here. A reason for usefulness of the Hamilton-Jacobi formulation is that
the parametrization by arc-length, which is incorporated in the ordinary differential equations describing
geodesics, is removed here.
· · ·
· · ·
.. .
· · ·
∂ψm
∂x1
∂ψm
∂x2
.
∂ψm
∂xm
⎞
⎟
⎠ .