## 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*.

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