ON THE DARBOUX EQUATION

Zbigniew Błocki

Abstract. Given a Riemannian surface M with a metric tensor g, we compute the Gauss curvature of a metric g − du ⊗ du, where u is a smooth function on M.

Introduction

In his celebrated monograph [2], Darboux, among other things, considered the problem of embedding abstract Riemannian surfaces in ℝ^{3}. If g is a metric tensor on M then one looks for three functions u, v, w on M, such that

g = du ⊗ du + dv ⊗ dv + dw ⊗ dw.

Locally, two of them, say v and w, must satisfy

g̃:= dv ⊗ dv + dw ⊗ dw > 0.

For the Gauss curvature K̃ of the new metric g̃ we thus have

(0.1)  K̃ = 0,

which is in fact an equation just for the first component u, the so called Darboux equation. Tedious calculations (see e.g. [3]) show that this equation, is, in modern terms, equivalent to

(0.2)  Mu = K(1 − |∇u|^{2}),

where M is the Monge-Ampère operator and K the Gauss curvature (with respect to the original metric g).

The aim of this note is to give the precise formula for K̃, which will show in particular that (0.1) and (0.2) are equivalent. We shall namely prove the following result.

Theorem. Suppose that M is a Riemannian manifold with metric tensor g and dim M = 2. For u ∈ C^{∞}(M) let

g̃ = g − du ⊗ du.

Then g̃ > 0 if and only if |∇u| < 1. In such a case the Gauss curvature with respect to g̃ is given by

(0.3)  K̃ = \frac{K(1 − |∇u|^{2}) − Mu}{(1 − |∇u|^{2})^{2}}.

Typeset by AMS-TEX

2000 Mathematics Subject Classification. 53C42.

Supported by KBN Grant 7 T07A 003 16.

1. Preliminaries

Here we collect the basic definitions and some formulas, which we will use in the proof of the theorem. For details we refer for example to [1]. Let M be a Riemannian manifold with the metric tensor g = ⟨·, ·⟩. g induces the unique symmetric, metric connection ∇ on M. This means that ∇ satisfies

(1.1)  ∇_{X}Y = ∇_{Y} X + [X, Y], X, Y ∈ X (M)

and ∇g = 0, that is

(1.2)  X⟨Y, Z⟩ − ⟨∇_{X}Y, Z⟩ − ⟨Y, ∇_{X}Z⟩ = 0, X, Y, Z ∈ X (M).

(1.1) and (1.2) are equivalent to

(1.3) \begin{matrix}2⟨∇_{X}Y, Z⟩ = & X⟨Y, Z⟩ + Y ⟨Z, X⟩ − Z⟨X, Y ⟩ \\ + ⟨[X, Y], Z⟩ + ⟨[Z, X], Y ⟩ − ⟨[Y, Z], X⟩, X, Y, Z ∈ X (M). \end{matrix}

In particular, we may compute

(1.4)  ⟨∇_{X}X, Y ⟩ = \frac{1}{2}Y |X|^{2} = ⟨∇_{Y} X, X⟩, X, Y ∈ X (M).

If u ∈ C^{∞}(M) then ∇u ∈ X (M) is uniquely defined by

⟨∇u, X⟩ = Xu, X ∈ X (M).

From (1.4) it follows that

(1.5)  ⟨∇_{∇u}∇u, X⟩ = \frac{1}{2}X|∇u|^{2} = \frac{1}{2}X∇u u = \frac{1}{2}⟨X∇u, ∇u⟩, X ∈ X (M).

Set

∇^{2}u : X (M) ∋ X ⟼ ∇_{X}∇u ∈ X (M).

Then ∇^{2}u is a C^{∞}(M)-linear endomorphism of the C^{∞}(M)-module X (M). The Monge-Ampere operator is defined by

Mu = det ∇^{2}u.

On M we have the Riemannian curvature tensor

R(X, Y; W, Z) = ⟨∇_{X}∇_{Y} Z − ∇_{Y} ∇_{X}Z − ∇_{[X,Y]}Z, W ⟩, X, Y, W, Z ∈ X (M).

If dim M = 2, the Gauss curvature of M is defined by

K = \frac{R(X, Y; X, Y)}{|X|^{2}|Y |^{2} − ⟨X, Y ⟩^{2}} ,

provided that, at a given point, X, Y ∈ X (M) span the tangent space.

2. Proof of the theorem

We have

g̃(X, Y) = ⟨X, Y ⟩ = ⟨X, Y ⟩ − Xu Y u, X, Y ∈ X (M).

Thus

(2.1)  g̃(∇u, ∇u) = (1 − |∇u|^{2})|∇u|^{2}.

If g̃ > 0 (i.e. g̃(X, X) > 0 for X ≠ 0), then it follows that |∇u| < 1. On the other hand, if |∇u| < 1 then for X ≠ 0 we get

g̃(X, X) = |X|^{2} − ⟨∇u, X⟩^{2} > 0

by the Schwartz inequality and (2.1). This proves the first statement.

Now assume that |∇u| < 1. On the interior of the set {∇u ≠ 0} (0.3) is clear. Since the result is purely local, and because both sides of (0.3) belong to C^{∞}(M), we may assume that ∇u ≠ 0 everywhere on M. We may also assume that there is W ∈ X (M) with W ≠ 0 and W u = 0 (in local coordinates we may choose W = (∂u/∂x_{2})∂_{1} − (∂u/∂x_{1})∂_{2}). This means that

(2.2)  ⟨∇u, W ⟩ = ⟨∇u, W ⟩ = 0.

Since ∇u/|∇u|, W/|W | form an orthonormal basis of the tangent space, we have

(2.3)  \begin{aligned}Mu & = \frac{⟨∇_{∇u}∇u, ∇u⟩⟨∇_{W} ∇u, W ⟩ − ⟨∇_{∇u}∇u, W ⟩⟨∇_{W} ∇u, ∇u⟩}{|∇u|^{2}|W |^{2}} \\ & = \frac{⟨∇_{∇u}∇u, ∇u⟩⟨∇_{W} ∇u, W ⟩ − ⟨∇_{∇u}∇u, W ⟩^{2}}{|∇u|^{2}|W |^{2}} , \end{aligned}

where the last equality follows from (1.4).

From (1.3) we obtain

⟨ ˜∇_{X}Y, Z⟩ = ⟨∇_{X}Y, Z⟩ − XY u Zu, X, Y, Z ∈ X (M).

This, (2.1) and (2.2) give

(2.4)  \begin{aligned}˜∇_{X}Y − ∇_{X}Y & = (\frac{⟨ ˜∇_{X}Y, ∇u⟩}{(1 − |∇u|^{2})|∇u|^{2}} − \frac{⟨∇_{X}Y, ∇u⟩}{|∇u|^{2}}) ∇u \\ & + (\frac{⟨ ˜∇_{X}Y, W ⟩}{|W |^{2}}− \frac{⟨∇_{X}Y, W ⟩}{|W |^{2}}) W \\ & = \frac{⟨∇_{X}Y − XY, ∇u⟩}{1 − |∇u|^{2}}∇u, X, Y ∈ X (M). \end{aligned}

In particular, by (1.5),

˜∇_{X}∇u − ∇_{X}∇u = −\frac{⟨∇_{∇u}∇u, X⟩}{1 − |∇u|^{2}}∇u, X ∈ X (M).

Hence, by (2.4), (2.2) and (2.3),

R̃(W, ∇u; W, ∇u) − R(W, ∇u; W, ∇u)

= ⟨∇_{W} ( ˜∇_{∇u}∇u − ∇_{∇u}∇u) − ∇_{∇u}( ˜∇_{W} ∇u − ∇_{W} ∇u), W ⟩

− \frac{⟨∇_{∇u}∇u, ∇u⟩}{1 − |∇u|^{2}}⟨∇_{W} ∇u, W ⟩ + \frac{⟨∇_{∇u}∇u, W ⟩}{1 − |∇u|^{2}}⟨∇_{∇u}∇u, W ⟩

= −\frac{|∇u|^{2}|W |^{2}Mu}{1 − |∇u|^{2}}.

This, together with (2.1), easily gives (0.3). □

References

[1] M. do Carmo, Riemannian Geometry, Birkkäuser, 1992.

[2] G. Darboux, Leçons sur la theorie générale des surfaces, vol. III, Gauthier-Villars, 1894.

[3] M. Spivak, A Comprehensive Introduction to Differential Geometry, vol. V, Publish or Perish, 1979.

Institute of Mathematics, Jagiellonian University, Reymonta 4, 30-059 Kraków, Poland

E-mail address: blocki@ im.uj.edu.pl