2
$\begingroup$

I am dealing with Riemannian immersions and I am stuck on the following: Given a totally geodesic immerserd surface $S$ on a compact riemannian manifold $M$ with metric $g$, is there another metric $\tilde g \neq g$ on $M$ such that $S$ is isometric immersed (not necessarely totally geodesically) on $(M,\tilde g)$? In this case, what can I say about the second fundamental form of this second immersion? Can I conclude with assumptions that the induced connection on $S$ is the same on both immersions?

$\endgroup$
1
  • 3
    $\begingroup$ If the connection on $M$ is the Levi-Civita connection, then the induced connection of an immersion is the Levi-Civita connection for the induced metric. Thus in both cases, since the immersion is isometric, the induced connection is the same. $\endgroup$ – Paul Bryan Oct 21 '17 at 1:44
2
$\begingroup$

Suppose that $S$ is a pair of disjoint Euclidean 2-spheres, say of radius 1 and radius 2. Isometrically immerse to the 3-sphere $M$ as totally geodesic spheres of radius 1, intersecting along a totally geodesic curve $C$ lying on each of the 2-spheres. The preimage of $C$ in $S$ is a pair of circles of different lengths. No perturbation of the metric on $M$ can give that curve $C$ those two different lengths. So $S$ is not isometrically immersed for any metric on $M$.

Suppose that you do have two immersions, one totally geodesic, the other isometric, $S \to M$. If they agree at one point, and have the same Levi-Civita connection induced, they are both isometric immersions. So we can easily invent a counterexample. Take $S$ the plane with standard metric, immersed into 3-dimensional Euclidean space by an immersion which is isometric only at the origin, and which puts $S$ into the horizontal 2-plane, so totally geodesic. Then the Levi--Civita connection of the pullback metric on $S$ cannot be the Levi--Civita connection of the original metric on $S$.

$\endgroup$

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.