## 3-Manifolds

Download zipped PostScript files of these joint papers with L. A. Cordero:

• Symmetries of Sectional Curvature on 3-Manifolds Demonstratio Math. 28 (1995) 635--650. [also available as a PDF]
• In dimension three, there are only two signatures of metric tensors: Lorentzian and Riemannian. We find the possible pointwise symmetry groups of Lorentzian sectional curvatures considered as rational functions, and determine which can be realized on naturally reductive homogeneous spaces. We also give some examples.

• Examples of Sectional Curvature with Prescribed Symmetry on 3-Manifolds Czech. Math. J. 45 (1995) 7--20. [PDF]
• In a previous paper, we determined the possible pointwise symmetry groups of sectional curvature considered as a rational function. We determined the naturally reductive homogeneous spaces with constant symmetry, and gave general descriptions of some examples of them. Here, we exhibit explicit forms of the metric tensors on some of these examples. We also give some inhomogeneous examples utilizing warped products, and begin the study of how the symmetry type can vary on a connected space.

• Left-invariant Lorentzian Metrics on 3-Dimensional Lie Groups Rend. Mat. Appl. 17 (1997) 129--155. [PDF]
• We find the Riemann curvature tensors of all left-invariant Lorentzian metrics on 3-dimensional Lie groups.

### Geodesic Surfaces in the Heisenberg Group

The following page shows two surfaces comprised of all the "equatorial" geodesics emanating from the identity element in the 3-dimensional Heisenberg group. The first picture is with a Lorentzian metric of signature (- - +), the second is with (+ - -), and the third is with (+ + -). To see "inside" the null cone, the third is rotated relative to the other two.

Geodesic Surfaces

In both pictures, the curves radiating or emanating outward from the origin (0,0,0) are geodesics and the "concentric circles" transverse to them are curves of constant t -value, with geodesic parameter t.