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 $\lambda$?


Thanks.



Eigenfunctions restricted on closed geodesics
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.


