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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.

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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

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  • $\begingroup$ 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, 2016 at 18:36
  • $\begingroup$ @AsafShachar I have edited the answer to include the case they have the same dimension. $\endgroup$
    – mfl
    Feb 6, 2016 at 18:45

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