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Consider the flat torus $T^2=\frac{\mathbb{R}^2}{l_1\mathbb{Z}\oplus l_2\mathbb{Z}}$. It is easy to see that the eigenvalues of the Laplacian on torus, $-\frac{\partial^2}{\partial x^2}-\frac{\partial^2}{\partial y^2}$, are $\lambda_{m_1,m_2}=(2\pi)^2(\frac{m_1^2}{l_1^2}+\frac{m_2^2}{l_2^2})$ with the associated eigenfunction $$f_{(m_1,m_2)}(x,y)=e^{2\pi i(\frac{m_1}{l_1}x+\frac{m_2}{l_2}y)}.$$ where $m_1,m_2\in \mathbb{Z}$. Furthermore, The closed geodesics of $T^2$ parametrized by the arc length, are $$ \gamma_{(n_1,n_2)}(t)=\frac{1}{l}(n_1l_1t,n_2l_2t)$$ where $n_1,n_2\in \mathbb{Z}$ and $l=\sqrt{n_1^2l_1^2+n_2^2l_2^2}$. A simple computation shows that an eigenfunction, say $f_{(m_1,m_2)}$, restricted on a closed geodesic, $\gamma_{(n_1,n_2)}$, gives
$$f_{(m_1,m_2)}\circ \gamma_{(n_1,n_2)}(t)=e^{2\pi i(\frac{m_1n_1+m_2n_2}{l})t}$$ Which is an eigenfunction on the circle $\mathbb{R}/l\mathbb{Z}$ with the eigenvalue $\tilde{\lambda}=\left( \frac{2\pi}{l}(m_1n_1+m_2n_2)\right)^2$.

Now my question is: Is this true in the general cases? More precisely;

Let $\gamma:[0,l]\to M$ be a closed geodesics on the Riemannian manifold $(M,g)$ which is parametrized by the arc length. If $f\in C^\infty(M)$ is an eigenfunction for the Laplacian on $M$, i.e. $$\Delta(f)=\lambda f$$ Then

Question 1) Is $f\circ \gamma$ an eigenfunction on the circle $S^1=\mathbb{R}/l\mathbb{Z}$? Or, Is it in the form of $$f\circ \gamma(t)=c e^{2\pi i \tilde{\lambda}t}.$$

Question 2) If so, how does $\tilde{\lambda}$ depend on $\gamma$ and $\lambda$?


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This is just a complement to Robert's answer: restrictions of eigenfunctions to geodesics (with a view to $L_p$ estimates) are treated here: Restrictions of the Laplace-Beltrami eigenfunctions to submanifolds N. Burq, P. Gérard, and N. Tzvetkov Source: Duke Math. J. Volume 138, Number 3 (2007), 445-486. – alvarezpaiva Jan 21 '13 at 8:42
up vote 11 down vote accepted

The answer is 'no', as you can see by taking the case of $M$ being the unit $2$-sphere in $\mathbb{R}^3$ and the geodesic $\gamma$ being a great circle, say, the horizontal great circle given by $z=0$. If you consider the harmonic polynomials of degree $2$ in $x,y,z$ restricted to the $2$-sphere, these are eigenfunctions of the Laplacian on the $2$-sphere, but their restrictions to the horizontal great circle aren't usually eigenfunctions of the Laplacian on the circle.

More generally, you take the $k$-th eigenspace of the Laplacian on the $2$-sphere for $k>1$, you'll find that the restriction of these functions to each great circle projects into a sum of a finite number of eigenspaces of the Laplacian on the great circle (I think it's about $\tfrac12(k{+}2)$ of them), but not into a single one of these eigenspaces.

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Thanks. You are right. – Asghar Ghorbanpour Jan 21 '13 at 15:26

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