I would like to understand the first section of a paper titled: "More on Convolution of Riemannian Manifolds," by Bang-Yen Chen. Michigan State University.
http://emis.impa.br/EMIS/journals/BAG/vol.44/no.1/b44h1chn.pdf (first two pages).
The notion of convolution products is defined as follows: Let $N_1$ and $N_2$ be two Riemannian manifolds equipped with metrics $g_1$ and $g_2,$ respectively. Consider the symmetric tensor field $g_{f,h}$ of type $(0,2)$ on the product manifold $N_1 × N_2$ defined by:
$$ g_{f,h} = h^2g_1 + f^2g_2 + 2fhdf ⊗ dh. $$
The author goes on to further develop the notion of a convolution of two Riemannian manifolds.
I tried an example:
$g_{f,h}=(x^4)g_1+(1-x)^4g_2+2(x^2)df ⊗ dh.$
How do I finish the computation? I'm not sure what to sub in for the metrics and I don't know how to deal with that last part, $df\otimes dh.$
Best Answer
This is supposed to be a generalization of warped metrics. Your attempt is not clear if you do not say what $N_1$ and $N_2$ are. Moreover, $f$ and $h$ are functions defined on the different factors, so it would not make sense for $x$ to appear multiplying both $g_1$ and $g_2$ (if I'm guessing right what you want to write). Also, $f$ and $h$ cannot vanish, but both $x^2$ and $(1-x)^2$ vanish at some point. I will attempt to clean up what you tried to do.
So, take $(N_1,g_1) = (\Bbb R_{>0}, {\rm d}x^2)$, $(N_2,g_2) = (\Bbb R, {\rm d}y^2)$. Take $f\colon \Bbb R_{>0} \to \Bbb R$ given by $f(x) = x^2$ and let me change your $h$: take $h\colon \Bbb R \to \Bbb R$ given by $h(y) = e^y$. Then the convolution metric on the manifold $\Bbb R_{>0}\times \Bbb R$ will be given by $$\begin{align} g_{f,h} &= e^{2y}\,{\rm d}x^2 + x^4\,{\rm d}y^2 + 2x^2e^y\,{\rm d}(x^2)\otimes {\rm d}(e^y) \\ &= e^{2y}\,{\rm d}x^2 + x^4\,{\rm d}y^2 + 2x^2e^y (2x\,{\rm d}x)\otimes (e^y\,{\rm d}y) \\ &= e^{2y}\,{\rm d}x^2 + x^4\,{\rm d}y^2 + 4x^3e^{2y}{\rm d}x\otimes {\rm d}y.\end{align}$$