## Hiroshima Mathematical Journal

### Infinitesimal isometries on tangent sphere bundles over three-dimensional manifolds

Tatsuo Konno

#### Abstract

In this article, we study the infinitesimal isometries on tangent sphere bundles over orientable three-dimensional Riemannian manifolds. Focusing on the vector fields which do not preserve fibers, we show the existence of lifts to the bundles of orthonormal frames. These lifts enable us to analyze the infinitesimal isometries by the symmetry of principal fiber bundles. We prove that the tangent sphere bundle admits a non-fiber-preserving infinitesimal isometry if and only if the base manifold has the same constant sectional curvatures as the fibers have. As an application, we classify the infinitesimal isometries on tangent sphere bundles for the three dimensional case.

#### Article information

Source
Hiroshima Math. J., Volume 41, Number 3 (2011), 343-366.

Dates
First available in Project Euclid: 12 December 2011

https://projecteuclid.org/euclid.hmj/1323700039

Digital Object Identifier
doi:10.32917/hmj/1323700039

Mathematical Reviews number (MathSciNet)
MR2895285

Zentralblatt MATH identifier
1235.53051

Subjects
Primary: 53C25: Special Riemannian manifolds (Einstein, Sasakian, etc.)
Secondary: 53C10: $G$-structures

#### Citation

Konno, Tatsuo. Infinitesimal isometries on tangent sphere bundles over three-dimensional manifolds. Hiroshima Math. J. 41 (2011), no. 3, 343--366. doi:10.32917/hmj/1323700039. https://projecteuclid.org/euclid.hmj/1323700039

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