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Let us insist that special loci, which are the fixed-point sets of the action of the modular group are simultaneously algebraic, arithmetic andtotallygeodesic.
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As we have said, we restrict ourselves to smooth timeliketotallygeodesicfoliations.
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There are many examples in the literature of perturbations of metrics preserving certain prescribedtotallygeodesicsubsets.
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Beware of the fact that these loci are nottotallygeodesicin general, except in the one-dimensional case.
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The integral manifolds of these distributions aretotallygeodesicsubmanifolds.
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A section is necessarilytotallygeodesic.
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This is equivalent to saying that the submanifold can be madetotallygeodesicby an appropriate conformal change of the metric of the surrounding (ambient) manifold.
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This example is from Wikipedia and may be reused under a CC BY-SA license.
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