# Classification of Hamilton-Jacobi separation in orthogonal coordinates with diagonal curvature

## Abstract

We find all orthogonal metrics where the geodesic Hamilton-Jacobi equation separates and the Riemann curvature tensor satisfies a certain equation (called the diagonal curvature condition). All orthogonal metrics of constant curvature satisfy the diagonal curvature condition. The metrics we find either correspond to a Benenti system or are warped product metrics where the induced metric on the base manifold corresponds to a Benenti system. Furthermore, we show that most metrics we find are characterized by concircular tensors; these metrics, called Kalnins-Eisenhart-Miller metrics, have an intrinsic characterization which can be used to obtain them on a given space. In conjunction with other results, we show that the metrics we found constitute all separable metrics for Riemannian spaces of constant curvature and de Sitter space.

- Authors:

- Department of Applied Mathematics, University of Waterloo, Waterloo, Ontario N2L 3G1 (Canada)

- Publication Date:

- OSTI Identifier:
- 22306029

- Resource Type:
- Journal Article

- Resource Relation:
- Journal Name: Journal of Mathematical Physics; Journal Volume: 55; Journal Issue: 8; Other Information: (c) 2014 AIP Publishing LLC; Country of input: International Atomic Energy Agency (IAEA)

- Country of Publication:
- United States

- Language:
- English

- Subject:
- 71 CLASSICAL AND QUANTUM MECHANICS, GENERAL PHYSICS; CLASSIFICATION; COORDINATES; DE SITTER SPACE; HAMILTON-JACOBI EQUATIONS; METRICS; RIEMANN SPACE; TENSORS

### Citation Formats

```
Rajaratnam, Krishan, E-mail: k2rajara@uwaterloo.ca, and McLenaghan, Raymond G., E-mail: rgmclenaghan@uwaterloo.ca.
```*Classification of Hamilton-Jacobi separation in orthogonal coordinates with diagonal curvature*. United States: N. p., 2014.
Web. doi:10.1063/1.4893335.

```
Rajaratnam, Krishan, E-mail: k2rajara@uwaterloo.ca, & McLenaghan, Raymond G., E-mail: rgmclenaghan@uwaterloo.ca.
```*Classification of Hamilton-Jacobi separation in orthogonal coordinates with diagonal curvature*. United States. doi:10.1063/1.4893335.

```
Rajaratnam, Krishan, E-mail: k2rajara@uwaterloo.ca, and McLenaghan, Raymond G., E-mail: rgmclenaghan@uwaterloo.ca. Fri .
"Classification of Hamilton-Jacobi separation in orthogonal coordinates with diagonal curvature". United States.
doi:10.1063/1.4893335.
```

```
@article{osti_22306029,
```

title = {Classification of Hamilton-Jacobi separation in orthogonal coordinates with diagonal curvature},

author = {Rajaratnam, Krishan, E-mail: k2rajara@uwaterloo.ca and McLenaghan, Raymond G., E-mail: rgmclenaghan@uwaterloo.ca},

abstractNote = {We find all orthogonal metrics where the geodesic Hamilton-Jacobi equation separates and the Riemann curvature tensor satisfies a certain equation (called the diagonal curvature condition). All orthogonal metrics of constant curvature satisfy the diagonal curvature condition. The metrics we find either correspond to a Benenti system or are warped product metrics where the induced metric on the base manifold corresponds to a Benenti system. Furthermore, we show that most metrics we find are characterized by concircular tensors; these metrics, called Kalnins-Eisenhart-Miller metrics, have an intrinsic characterization which can be used to obtain them on a given space. In conjunction with other results, we show that the metrics we found constitute all separable metrics for Riemannian spaces of constant curvature and de Sitter space.},

doi = {10.1063/1.4893335},

journal = {Journal of Mathematical Physics},

number = 8,

volume = 55,

place = {United States},

year = {Fri Aug 15 00:00:00 EDT 2014},

month = {Fri Aug 15 00:00:00 EDT 2014}

}