Random 3-manifolds have no totally geodesic submanifolds

Abstract Murphy and the second author showed that a generic closed Riemannian manifold has no totally geodesic submanifolds, provided the ambient space is at least four dimensional. Lytchak and Petrunin established a similar result in dimension 3. For the higher dimensional result, the “generic set” is open and dense in the $$C^{q}$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msup> <mml:mi>C</mml:mi> <mml:mi>q</mml:mi> </mml:msup> </mml:math> –topology for any $$q\ge 2.$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>q</mml:mi> <mml:mo>≥</mml:mo> <mml:mn>2</mml:mn> <mml:mo>.</mml:mo> </mml:mrow> </mml:math> In Lytchak and Petrunin’s work, the “generic set” is a dense $$G_{\delta }$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msub> <mml:mi>G</mml:mi> <mml:mi>δ</mml:mi> </mml:msub> </mml:math> in the $$C^{q}$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msup> <mml:mi>C</mml:mi> <mml:mi>q</mml:mi> </mml:msup> </mml:math> –topology for any $$q\ge 2.$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>q</mml:mi> <mml:mo>≥</mml:mo> <mml:mn>2</mml:mn> <mml:mo>.</mml:mo> </mml:mrow> </mml:math> Here we show that the set of such metrics on a compact 3–manifold actually contains a set that is that is open and dense set in the $$C^{q}$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msup> <mml:mi>C</mml:mi> <mml:mi>q</mml:mi> </mml:msup> </mml:math> –topology, provided $$q\ge 3.$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>q</mml:mi> <mml:mo>≥</mml:mo> <mml:mn>3</mml:mn> <mml:mo>.</mml:mo> </mml:mrow> </mml:math>