##
**More on convolution of Riemannian manifolds.**
*(English)*
Zbl 1031.53099

Let \((M_1,g_1)\) and \((M_2,g_2)\) be two Riemannian manifolds. Consider \(M_1\times M_2\) and the symmetric tensor field \(g_{f,h}\) of type \((0,2)\) on it defined by
\[
g_{f,h}= h^2g_1+ f^2g_2+f h(df\otimes dh+dh \otimes df)
\]
for some positive differentiable functions \(f\) and \(h\) on \(M_1\) and \(M_2\), respectively. \(g_{f,h}\) (denoted by \(_hg_1* _fg_2)\) is called the convolution of \(g_1\) and \(g_2\) via \(h\) and \(f\). \(M_1\times M_2\) equipped with \(_hg_1* _fg_2\) is called a convolution manifold, denoted by \(_hM_1* _gM_2\). When this metric is nondegenerate, it defines a pseudo-Riemannian metric on \(M_1\times M_2\) with index \(\leq 1\) and then \(_hM_1* _fM_2\) is said to be a convolution pseudo-Riemannian manifold. When the index is zero, it is called a convolution Riemannian manifold.

This kind of spaces and metrics have been introduced and studied by B. Y. Chen in [Bull. Aust. Math. Soc. 66, 177-191 (2002; Zbl 1041.53012)] and the author continues this study in the paper under review. First he provides some new examples of convolution manifolds by using the tensor product of Euclidean submanifolds. Moreover, he derives the following necessary and sufficient condition for the convolution metric \(_hg_1* _fg_2\) of two Riemannian metrics to be a Riemannian metric: \(\|\text{grad} f\|_1 \|\text{grad} h\|_2 <1\) where \(\|\|\) denotes the length. The construction of the above mentioned examples leads the author to the construction of examples, and their study, of submanifolds in pseudo-Euclidean spaces whose distance function \(x\) satisfies \(\|\text{grad} x \|=c\in [0,\infty)\). Furthermore, he obtains a necessary and sufficient condition for one of the factors of a convolution Riemannian manifold to be totally geodesic and provides the complete list when the geodesic factor is an open interval of the real line. In the final section he completely classifies the flat convolution Riemannian surfaces.

This kind of spaces and metrics have been introduced and studied by B. Y. Chen in [Bull. Aust. Math. Soc. 66, 177-191 (2002; Zbl 1041.53012)] and the author continues this study in the paper under review. First he provides some new examples of convolution manifolds by using the tensor product of Euclidean submanifolds. Moreover, he derives the following necessary and sufficient condition for the convolution metric \(_hg_1* _fg_2\) of two Riemannian metrics to be a Riemannian metric: \(\|\text{grad} f\|_1 \|\text{grad} h\|_2 <1\) where \(\|\|\) denotes the length. The construction of the above mentioned examples leads the author to the construction of examples, and their study, of submanifolds in pseudo-Euclidean spaces whose distance function \(x\) satisfies \(\|\text{grad} x \|=c\in [0,\infty)\). Furthermore, he obtains a necessary and sufficient condition for one of the factors of a convolution Riemannian manifold to be totally geodesic and provides the complete list when the geodesic factor is an open interval of the real line. In the final section he completely classifies the flat convolution Riemannian surfaces.

Reviewer: L.Vanhecke (Leuven)

### MSC:

53C50 | Global differential geometry of Lorentz manifolds, manifolds with indefinite metrics |

53C20 | Global Riemannian geometry, including pinching |

53C40 | Global submanifolds |