Matrix Differential Calculus with Applications in Statistics and Econometrics

Oxford University Press in Africa, 1927â80

Linear operators part ii (spectral theory)

Manifolds all of whose geodesics are closed

https://www.kau.edu.sa/Files/130/Researches/60166_30987.pdf

On four-point nonlocal boundary value problems of nonlinear integro-differential equations of fractional order

We define the Floer complex for Hamiltonian orbits on the cotangent bundle of a compact manifold which satisfy non-local conormal boundary conditions. We prove that the homology of this chain complex is isomorphic to the singular homology of the natural path space associated to the boundary conditions.

https://www.mis.mpg.de/preprints/2008/preprint2008_61.pdf

The homology of path spaces and Floer homology with conormal boundary conditions

Perturbed geodesics are trajectories of particles moving on 
a semi-Riemannian manifold in the presence of a potential. Our
purpose here is to extend to perturbed geodesics on
semi-Riemannian manifolds the well known Morse Index Theorem. When
the metric is indefinite, the Morse index of the energy
functional becomes infinite and hence, in order to obtain a
meaningful statement, we substitute the Morse index by its
relative form, given by the spectral flow of an associated family
of index forms. We also introduce a new counting for conjugate
points, which need not to be isolated in this context, and prove
that our generalized Morse index equals the total number of
conjugate points. Finally we study the relation with the Maslov
index of the flow induced on the Lagrangian Grassmannian.

/pdf/a-morse-index-theorem-for-perturbed-geodesics-on-semi-36rgu79dsp.pdf

A Morse index theorem for perturbed geodesics on semi-Riemannian manifolds

We give a functional analytical proof of the equalitybetween the Maslov index of a semi-Riemannian geodesicand the spectral flow of the path of self-adjointFredholm operators obtained from the index form. This fact, together with recent results on the bifurcation for critical points of strongly indefinite functionals imply that each nondegenerate and nonnull conjugate (or P-focal)point along a semi-Riemannian geodesic is a bifurcation point.In particular, the semi-Riemannian exponential map is notinjective in any neighborhood of a nondegenerate conjugate point,extending a classical Riemannian result originally due to Morse and Littauer.

Spectral Flow, Maslov Index and Bifurcation of Semi-Riemannian Geodesics

Index theory revealed its outstanding role in the study of periodic orbits of Hamiltonian systems and the dynamical consequences of this theory are enormous. Although the index theory in the periodic case is well-established, very few results are known in the case of homoclinic orbits of Hamiltonian systems. Moreover, to the authors’ knowledge, no results have been yet proved in the case of heteroclinic and halfclinic (i.e. parametrized by a half-line) orbits. Motivated by the importance played by these motions in understanding several challenging problems in Classical Mechanics, we develop a new index theory and we prove at once a general spectral flow formula for heteroclinic, homoclinic and halfclinic trajectories. Finally we show how this index theory can be used to recover all the (classical) existing results on orbits parametrized by bounded intervals.

Index theory for heteroclinic orbits of Hamiltonian systems

http://calvino.polito.it/ricerca/2004/pdf/3_2004/art_3_2004.pdf

Alessandro Portaluri

Papers

The homology of path spaces and Floer homology with conormal boundary conditions

A Morse index theorem for perturbed geodesics on semi-Riemannian manifolds

Spectral Flow, Maslov Index and Bifurcation of Semi-Riemannian Geodesics

Index theory for heteroclinic orbits of Hamiltonian systems

Computation of the Maslov index and the spectral flow via partial signatures