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Could there be a realizable 2-surface in some higher dimensional non-Riemannian embedding space whose second fundamental form is skew? If yes, then what would its skew part mean?

Ayan
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    What exactly do you mean by a non-Riemannian embedding space? And by "skew," do you mean that you want the second fundamental form to be skew-symmetric? – Jesse Madnick Apr 14 '15 at 10:16
  • If you isometrically embed a Riemannian 2-manifold into a higher dimensional Riemannian manifold, then the second fundamental form, $\text{II}$, will be a symmetric bilinear form. If $\text{II}$ were also skew-symmetric, then it would be zero, meaning that the surface is "totally geodesic." The "totally geodesic" property has a few geometric interpretations. That said, I don't know what you mean by a "non-Riemannian embedding space." – Jesse Madnick Apr 14 '15 at 10:23
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    By non-Riemannian embedding space, I mean a 2-manifold being embedded in a space whose connection is not symmetric, and probably non-metric (i.e. it has a non-zero torsion tensor and the covariant derivative of the metric is non-zero). The connection and metric that the embedding space on the 2-manifold would not be the usual symmetric and metric connection then. In that case, could the second fundamental form be non-symmetric? If yes, then what would its skew part geometrically signify? – Ayan Apr 14 '15 at 11:10

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