# zbMATH — the first resource for mathematics

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

##### 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
Full Text: