2
$\begingroup$

Let $(M,g),(N,h)$ be two Riemannian manifolds.

Assume $f:M \to N$ is an isometric immersion. Is $f$ harmonic? (i.e a critical point of the energy functional)

(I know this is true when $(N,h)=(\mathbb{R}^n,e)$ is the standard Euclidean space).

Edit: I assume also $M$,$N$ are manifolds of the same dimension.

$\endgroup$
2
$\begingroup$

It is not true, even if $N$ is the Euclidean space. Just consider the curve

$$\gamma: (0,2\pi)\to \mathbb{R}^2, \quad t \to (\cos t, \sin t).$$ Since $|\gamma'(t)|= 1$ it is an isometric immersion. But it is not harmonic, because harmonic curves are geodesics. And, $\gamma$ is not a geodesic.

Edit

In case $\dim M=\dim N$ the answer is yes. See the bottom of the $5^{th}$ page in http://www1.maths.leeds.ac.uk/pure/staff/wood/Hamaps-hbk.pdf

$\endgroup$
2
  • $\begingroup$ You are right. I forgot to mention I am only interested in the case where $\dim M =\dim N$. Do you know if it holds in that case? $\endgroup$ Feb 6 '16 at 18:36
  • $\begingroup$ @AsafShachar I have edited the answer to include the case they have the same dimension. $\endgroup$
    – mfl
    Feb 6 '16 at 18:45

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.