In the present paper, we give an invariant on isometric immersions into spaces of constant sectional curvature. This invariant is a direct consequence of the Gauss equation and the Codazzi equation of isometric immersions. We apply this invariant on some examples. Further, we apply it to codimension 1 local isometric immersions of 2-step nilpotent Lie groups with arbitrary leftinvariant Riemannian metric into spaces of constant nonpositive sectional curvature. We also consider the more general class, namely, three-dimensional Lie groups G with nontrivial center and with arbitrary left-invariant metric. We show that if the metric of G is not symmetric, then there are no local isometric immersions of G into Q c 4.
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Published in Ukrains’kyi Matematychnyi Zhurnal, Vol. 61, No. 12, pp. 1660–1667, December, 2009.
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Rivertz, H.J. On an invariant on isometric immersions into spaces of constant sectional curvature. Ukr Math J 61, 1946–1955 (2009). https://doi.org/10.1007/s11253-010-0324-8
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DOI: https://doi.org/10.1007/s11253-010-0324-8