Complete classification of parallel Lorentz surfaces in neutral pseudo hyperbolic 4-space |
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Authors: | Bang-Yen Chen |
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Institution: | 1.Department of Mathematics,Michigan State University,East Lansing,USA |
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Abstract: | A Lorentz surface of an indefinite space form is called a parallel surface if its second fundamental form is parallel with
respect to the Van der Waerden-Bortolotti connection. Such surfaces are locally invariant under the reflection with respect
to the normal space at each point. Parallel surfaces are important in geometry as well as in general relativity since extrinsic
invariants of such surfaces do not change from point to point. Recently, parallel Lorentz surfaces in 4D neutral pseudo Euclidean
4-space $
\mathbb{E}_2^4
$
\mathbb{E}_2^4
and in neutral pseudo 4-sphere S
24 (1) were classified in 14] and in 10], respectively. In this paper, we completely classify parallel Lorentz surfaces in
neutral pseudo hyperbolic 4-space H
24 (−1). Our main result states that there are 53 families of parallel Lorentz surfaces in H
24 (−1). Conversely, every parallel Lorentz surface in H
24 (−1) is obtained from the 53 families. As an immediate by-product, we achieve the complete classification of all parallel
Lorentz surfaces in 4D neutral indefinite space forms. |
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Keywords: | |
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