= Totally geodesic hypersurface
= Totally geodesic
{synonym}
A <hypersurface> is <totally geodesic> when its intrinsic <geodesics> are also ambient <geodesics>. Equivalently, for a non-null hypersurface with the induced metric, its <extrinsic curvature> vanishes. The normal part of the ambient <covariant derivative along a curve> of the tangent is the <second fundamental form>, so its vanishing identifies the two <geodesic> equations.
This is a local statement: an ambient <geodesic> starting tangent to the hypersurface stays in it for a sufficiently short interval. Global containment needs the relevant completeness or closedness hypothesis. An open line segment inside the Euclidean plane has zero <extrinsic curvature>, but an ambient straight line leaves the segment in finite time. The whole line is <totally geodesic> and complete. A great circle on a round sphere is another complete example.
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