Geodesic
Differentiable functions in associative and alternative algebras and smooth surfaces in projective spaces over these algebras
Publications de l'Institut Mathématique, _N_S_68 (2000) no. 82, p. 67

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The following assertions are proved: (1) in simple noncommutative associative and alternative algebras only linear functions $y=ax+b$ and $y=xa+b$ have left and right derivatives, and (2) in the space over all commutative associative algebras smooth $m$-surfaces (lines for $m=1$) have tangent $m$-planes depending on the same number of parameters as points in surfaces. In the spaces over simple noncommutative associative and alternative algebras only $m$-planes (straight lines for $m=1$) are smooth $m$-surfaces. In the spaces over nonsemisimple noncommutative algebras smooth $m$-surfaces have tangent $m$-planes depending on the number of parameters less than points in surfaces.
Classification : 17A99 53A40 
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     author = {Boris A. Rosenfeld},
     title = {Differentiable functions in associative and alternative algebras and smooth surfaces in projective spaces over these algebras},
     journal = {Publications de l'Institut Math\'ematique},
     pages = {67 },
     publisher = {mathdoc},
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     number = {82},
     year = {2000},
     zbl = {1058.53013},
     language = {en},
     url = {http://geodesic.mathdoc.fr/item/PIM_2000_N_S_68_82_a6/}
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Boris A. Rosenfeld. Differentiable functions in associative and alternative algebras and smooth surfaces in projective spaces over these algebras. Publications de l'Institut Mathématique, _N_S_68 (2000) no. 82, p. 67 . http://geodesic.mathdoc.fr/item/PIM_2000_N_S_68_82_a6/