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# The Splitting Problem in Riemannian and Lorentzian Geometry IV

## José Luís Flores Universidad de Málaga

In this brief course of Lectures on Lorentzian Geometry we are going to review the main ideas involved in the formulation and proof of the so called "Rieman- nian and Lorentzian Splitting Theorems". We will put special emphasis on the similarities and differences in the arguments when passing from the Riemannian to the Lorentzian case, and the way in which these obstacles are overcome in the proofs. We will begin by establishing the Riemannian Splitting Theorem by Cheeger and Gromoll, including some comments about the initial motivation and its precedents. Next, we will give some basic notions and previous results useful for the proof of the theorem. Then, we will outline the main ideas of the original proof due to Cheeger and Gromoll. We will also study an alternative proof provided by Eschenburg and Heintze, which minimizes the use of elliptic theory, and thus, becomes very useful in the Lorentzian version of the theorem. Next, we will consider the Lorentizian case. After recalling some basic no- tions and results of this geometry, we will establish the Lorentzian Splitting Theorem, including some comments about the main hits in the history of its proof. For the proof of this result, which will be described with certain detail, we will essentially follow the approach by Galloway in [J. Diff. Geom. 29 (1989), 373-387]. Finally, we will conclude the course by analyzing some open problems related to these theorems.

# The Splitting Problem in Riemannian and Lorentzian Geometry III

## José Luís Flores Universidad de Málaga

In this brief course of Lectures on Lorentzian Geometry we are going to review the main ideas involved in the formulation and proof of the so called "Rieman- nian and Lorentzian Splitting Theorems". We will put special emphasis on the similarities and differences in the arguments when passing from the Riemannian to the Lorentzian case, and the way in which these obstacles are overcome in the proofs. We will begin by establishing the Riemannian Splitting Theorem by Cheeger and Gromoll, including some comments about the initial motivation and its precedents. Next, we will give some basic notions and previous results useful for the proof of the theorem. Then, we will outline the main ideas of the original proof due to Cheeger and Gromoll. We will also study an alternative proof provided by Eschenburg and Heintze, which minimizes the use of elliptic theory, and thus, becomes very useful in the Lorentzian version of the theorem. Next, we will consider the Lorentizian case. After recalling some basic no- tions and results of this geometry, we will establish the Lorentzian Splitting Theorem, including some comments about the main hits in the history of its proof. For the proof of this result, which will be described with certain detail, we will essentially follow the approach by Galloway in [J. Diff. Geom. 29 (1989), 373-387]. Finally, we will conclude the course by analyzing some open problems related to these theorems.

# The Splitting Problem in Riemannian and Lorentzian Geometry II

## José Luís Flores Universidad de Málaga

In this brief course of Lectures on Lorentzian Geometry we are going to review the main ideas involved in the formulation and proof of the so called "Rieman- nian and Lorentzian Splitting Theorems". We will put special emphasis on the similarities and differences in the arguments when passing from the Riemannian to the Lorentzian case, and the way in which these obstacles are overcome in the proofs. We will begin by establishing the Riemannian Splitting Theorem by Cheeger and Gromoll, including some comments about the initial motivation and its precedents. Next, we will give some basic notions and previous results useful for the proof of the theorem. Then, we will outline the main ideas of the original proof due to Cheeger and Gromoll. We will also study an alternative proof provided by Eschenburg and Heintze, which minimizes the use of elliptic theory, and thus, becomes very useful in the Lorentzian version of the theorem. Next, we will consider the Lorentizian case. After recalling some basic no- tions and results of this geometry, we will establish the Lorentzian Splitting Theorem, including some comments about the main hits in the history of its proof. For the proof of this result, which will be described with certain detail, we will essentially follow the approach by Galloway in [J. Diff. Geom. 29 (1989), 373-387]. Finally, we will conclude the course by analyzing some open problems related to these theorems.

# The Splitting Problem in Riemannian and Lorentzian Geometry

## José Luís Flores Universidad de Málaga

In this brief course of Lectures on Lorentzian Geometry we are going to review the main ideas involved in the formulation and proof of the so called "Rieman- nian and Lorentzian Splitting Theorems". We will put special emphasis on the similarities and differences in the arguments when passing from the Riemannian to the Lorentzian case, and the way in which these obstacles are overcome in the proofs. We will begin by establishing the Riemannian Splitting Theorem by Cheeger and Gromoll, including some comments about the initial motivation and its precedents. Next, we will give some basic notions and previous results useful for the proof of the theorem. Then, we will outline the main ideas of the original proof due to Cheeger and Gromoll. We will also study an alternative proof provided by Eschenburg and Heintze, which minimizes the use of elliptic theory, and thus, becomes very useful in the Lorentzian version of the theorem. Next, we will consider the Lorentizian case. After recalling some basic no- tions and results of this geometry, we will establish the Lorentzian Splitting Theorem, including some comments about the main hits in the history of its proof. For the proof of this result, which will be described with certain detail, we will essentially follow the approach by Galloway in [J. Diff. Geom. 29 (1989), 373-387]. Finally, we will conclude the course by analyzing some open problems related to these theorems.