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As indicated by the title, I am looking to find the Green's function for the Laplacian on $S^1 \times S^2$. Is such a function known? If not, does anyone have an approach to constructing such a function? My first idea isto combine the Green's function on $S^2$ in a nice way with some $1$-periodic function on $\mathbb{R}$, but I haven't had much luck.

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    $\begingroup$ Is it possible to do the following: an element in $S^1 \times S^2$ may be written $(x_1, x_2, x_3, x_4, x_5)$ with $x_1^2+x_2^2 =1$, $x_3^2+x_4^2+x_5^2 = 1$. Embed into $\mathbb{R}^5$ via the identity map, say $\varphi$. Then just compose with the Green's function on $\mathbb{R}^5$, i.e. take $G(\varphi(x),\varphi(x'))$. It just seems like a rather naive approach and I'm unsure of myself. $\endgroup$ – GiantTortoise1729 Jul 15 '16 at 14:35
  • $\begingroup$ What are you mean by construction? Would a representation as some unwieldy series or an integral do? $\endgroup$ – Andrew Jul 18 '16 at 15:02
  • $\begingroup$ Yes, that would suffice $\endgroup$ – GiantTortoise1729 Jul 18 '16 at 18:02
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Here's a way to get a representation as series. It works in a more general situation. Let $M$ and $N$ be smooth closed Riemannian manifolds. Denote by $\{\varphi_i\}_{i=1}^\infty$, $\{\psi_j\}_{i=j}^\infty$ the eigenfunctions with eigenvalues $\lambda_i$ and $\mu_j$ for the Laplace operators on $M$ and $N$ respectively. Let $L_2$ norms of eigenfunctions be equal to one so delta-functions can be expanded as $$ \delta_M(x-x')=\sum_{i=1}^\infty \varphi_i(x) \varphi_i(x'), $$ $$ \delta_N(y-y')=\sum_{i=1}^\infty \psi_i(y) \psi_i(y'). $$ Then Green's function for $M$ is given by series $$ G_M(x,x')=\sum_{i=1}^\infty \frac{\varphi_i(x) \varphi_i(x')}{\lambda_i} $$ and analogously for $N$. The Green's function for $M\times N$ is $$ G_{M\times N}(x,y,x',y')= \sum_{i,j=1}^\infty \frac{\varphi_i(x)\varphi_i(x') \psi_j(y)\psi_j(y')}{\lambda_i+\mu_j} $$ since $$ \Delta_{x,y} G_{M\times N}(x,y,x',y')= \sum_{i,j=1}^\infty \varphi_i(x)\varphi_i(x') \psi_j(y)\psi_j(y')= $$ $$ \delta_M(x-x')\delta_N(y-y')=\delta_{M\times N}(x-x',y-y'). $$ So one has to combine eigenfunctions rather than Green's functions themselves.

For your case $\varphi_i$ are cosines (up to a constant) and $\psi_j$ are spherical harmonics. A possibility to get a closed form for this series seems rather thin to me.

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  • $\begingroup$ Yes, getting a closed form does seem dubious. But thank you very much! $\endgroup$ – GiantTortoise1729 Jul 19 '16 at 15:14
  • $\begingroup$ I will wait to award the bounty. $\endgroup$ – GiantTortoise1729 Jul 19 '16 at 15:14
  • $\begingroup$ do you know a nice way of writing down these eigenfunctions? $\endgroup$ – GiantTortoise1729 Jul 21 '16 at 15:33
  • $\begingroup$ Nothing more nice that a standard definition en.wikipedia.org/wiki/Spherical_harmonics $\endgroup$ – Andrew Jul 21 '16 at 19:55
  • $\begingroup$ +1 very nice, any good books on this subject? $\endgroup$ – nik Jul 22 '16 at 14:32

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