A Morse complex for Lorentzian geodesics. (English) Zbl 1166.58009

Abbondandolo, Alberto; Majer, Pietro

Asian J. Math. 12, No. 3, 299-320 (2008).

Let \((M,h)\) be a Lorentzian manifold of finite dimension. If \(\gamma\) is a curve in \(M\), the energy functional is defined according to \(E(\gamma)=\frac12 \int_0^1 h(\gamma',\gamma')\). A geodesic is a critical point of \(E\), and unlike the Riemannian case, all critical points of \(E\) have infinite Morse index \(i(\gamma)\), so \(i\) is not a suitable invariant to classify all geodesics joining given endpoints. However, a finite relative index can be defined for geodesics in \(M\), and this index is known to be equal to the Maslov index of \(\gamma\).
The authors, using the Morse complex approach, prove Morse relations which involve the relative index, analogous to the Morse relations of Riemannian geometry [see for instance R. S. Palais, Topology 2, 299–340 (1963; Zbl 0122.10702)], for the set of all geodesics connecting two non-conjugate points in \(M\).

Reviewer: Gabriel Larotonda (Buenos Aires)

MSC:

58E10	Variational problems in applications to the theory of geodesics (problems in one independent variable)
53C50	Global differential geometry of Lorentz manifolds, manifolds with indefinite metrics

Keywords:

geodesic; Lorentzian manifold; Maslov index; Morse complex; Morse relations