Caloric morphisms for rotation invariant metrics (Q633866)

The authors characterize all the caloric morphism for rotation invariant metrics, when the space dimension is greater than two. Let \((M,g)\), \((N,h)\) be Riemannian manifolds, \(f\) a \(C^\infty\)-mapping from a domain \(D \subset \mathbb{R} \times M\) into \(\mathbb{R} \times N\) and \( \varphi\) a strictly positive \(C^\infty\)-function on \(D\). If the following properties hold {\parindent=6mm\begin{itemize}\item[1)] \(f(D)\) is a domain in \(\mathbb{R} \times N\); and \item[2)] the function \( \varphi (t,x)(u \circ f)(t,x)\) is caloric on \(f^{-1} (E)\), for any caloric function \(u\) defined on the open set \(E\) in \(\mathbb{R} \times N\) \end{itemize}} then the pair \((f, \varphi )\) is referred to as a caloric morphism defined on \(D\). Let \(g\) be a rotation invariant metric on \(M = \mathbb{R}^n \backslash \{0\}\) where \(n>2\). The main result of the article gives eight characterizations. That is, if \((f, \varphi )\) is a caloric morphism defined on \(D\), then one of the following cases occurs, up to isometry of \(M\), time translation, constant multiple of \(\varphi\) and the conjugate by isometry and time scaling: {\parindent=8mm\begin{itemize} \item[(A1)] \(g=|x|^q g_0\), where \(q\) is a positive constant, with \(q \neq -2\), and \(f(t,x)=(-1/a^2 t, x/ |at|^{2}/(q+2))\), \( \varphi (t,x)=\frac{1}{|t|^{ n/2}}\exp(-|x|^{q+2}/(q+2)^{2}t))\), with \(a\) a positive constant. \item [(A2)] For \(q=-2\), \(f(t,x)=(t,ce^{at} x)\) and \( \varphi (t,x)=|x|^{\frac{a}{2}}e^{\frac{a^2 t}{4}}\), where \(a,c\in \mathbb{R}\), with \(c\) positive. \item[(A3)] \(g=\rho(|x|)g_0\), where \(\rho (r)\) satisfies \(\rho (\nu r)= \nu^{q}\rho(r)\) and \(f (t,x)= (\nu^{q+2}t,\nu x), \varphi (t,x)= 1\), where \(\nu, q \in \mathbb{R}\) and \(\nu\) is positive. \item [(B1)] \(g=|x|^{-2}g_0\) and \(f (t,ce^{at}\frac{x}{|x|^{2}})\), \(\varphi (t,x)=\frac{e^{0.25a^{2}t}}{|x|^{ a/2}}\), where \(a,c\in \mathbb{R}\), with \(c\) a positive constant. \item [(B2)] \(g=\rho(|x|)g_0\), where \(\rho (r)\) satisfies \(\rho(\nu/r)= (\lambda r^{4}/\nu^{2})\rho (r)\), \(f (t,x)= (\lambda t,\nu x/|x|^{2})\), \(\varphi (t,x)= 1\), where \(\nu, \lambda\) are positive constants. \item [(C1)] \(g=\frac{g_0}{(|x|^{2}+1)}^{2}\) and \(f (t,x)=(t,x) , \varphi (t,x)= 1\). \item [(C2)] \(g=\frac{g_0}{(|x|^{2}-1)}^{2}\) and \(f(t,x)=(t,x), \varphi (t,x)= 1\). \item [(C3)] \(g=g_0\) and \item [(i)] \(f(t,x)= (-\frac{a^{2}}{t},\frac{ax}{t}), \varphi (t,x)= \frac{1}{(4\pi|t|)}^{\frac{n}{2}} \exp (-|x|^{2}/4t)\); or \item [(ii)] \(f(t,x)= (a^{2}t,a(x+tv)), \varphi (t,x)= \exp(\frac{|v|^{2}t}{4}+\frac{1}{2}v\cdot x)\), \end{itemize}} where \(a\neq0\) is a constant and \(v \in \mathbb{R}^{n}\), and \(v\cdot x\) is an inner product. Throughout \(g_0\) denotes the Euclidean metric on \(\mathbb{R}^{n}\). The result is also stated in radical metric form.