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In this paper we establish quaternionic and octonionic analogs of theclassical Riemann surfaces. The construction of these manifolds has nicepeculiarities and the scrutiny of Bernhard Riemann approach to Riemannsurfaces, mainly based on conformality, leads to the definition of sliceconformal or slice isothermal parameterization of quaternionic or octonionicRiemann manifolds. These new classes of manifolds include slice regularquaternionic and octonionic curves, graphs of slice regular functions, the $4$and $8$ dimensional spheres, the helicoidal and catenoidal $4$ and $8$dimensional manifolds. Using appropriate Riemann manifolds, we also give aunified definition of the quaternionic and octonionic logarithm and $n$-th rootfunction.

Slice conformality and Riemann manifolds on quaternions and octonions