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Vgl.M. Pinl, Monatshefte für Mathematik55/3 (1951), S. 188–199.
Vgl.M. Pinl, Monatshefte für Mathematik55/3 (1951), S. 195.
Da η (u)=r(u 1,u 2)+i ζ (u 1,u 2) inu analytisch, folgt nachCauchy Riemann \(\mathfrak{r}_1 = \mathfrak{s}_2 ,\mathfrak{r}_2 = - \mathfrak{s}_1 ,\Delta \mathfrak{r} = \mathfrak{r}_{11} + \mathfrak{r}_{22} = 0,\Delta \mathfrak{s} = \mathfrak{s}_{11} + \mathfrak{s}_{22} = 0\) und da η isotrop\(\mathfrak{h}'^2 = (\mathfrak{r}_1 + i\mathfrak{s}_1 )^2 = (\mathfrak{r}_1 - i\mathfrak{r}_2 )^2 = (i\mathfrak{s}_1 + \mathfrak{s}_2 )^2 = 0.\) Daher ist\(\mathfrak{r}_1^2 = \mathfrak{r}_2^2 = \lambda ,\mathfrak{r}_1 \mathfrak{r}_2 = 0bzw.g_{11} = g_{22} = \lambda (u_1 ,u_2 ) = |0,g_{12} = 0.\) Die Indizes der Vektoren bedeuten partielle Ableitungen, z. B.:\(\mathfrak{r}_1 = \frac{{\partial \mathfrak{r}}}{{\partial u_1 }},\mathfrak{r}_2 = \frac{{\partial \mathfrak{r}}}{{\partial u_2 }},\mathfrak{r}_{11} = \frac{{\partial ^2 \mathfrak{r}}}{{\partial u_{_1 }^2 }},\mathfrak{r}_{12} = \frac{{\partial ^2 \mathfrak{r}}}{{\partial u_1 \partial u_2 }},\mathfrak{r}_{22} = \frac{{\partial ^2 \mathfrak{r}}}{{\partial u_2^2 }},\mathfrak{s}_1 = \frac{{\partial \mathfrak{s}}}{{\partial u_1 }},\mathfrak{s}_2 = \frac{{\partial \mathfrak{s}}}{{\partial u_2 }},...\)
Vgl.W. Blaschke, Annali di Matematica pura ed applicata, Serie IV-Tomo XXVIII-1949, p. 205–209.
Vgl.M. Pinl, B-Kugelbilder der reellen Minimalflächen inR 4. Math. Z.59, S. 290–295 (1953).
Vgl. (5),, S. 294 (1953).
Vgl. (4),W. Blaschke, Annali di Matematica pura ed applicata, Serie IV-Tomo XXVIII-1949, p. 208.
Vgl. z. B.W. Blaschke, Differentialgeometrie, Band I, Springer-Verlag, Berlin; Comment. math. helv.4, 1932, S. 248–255.
Vgl.L. Bieberbach, Verhandlungen des internationalen Mathematiker-Kongresses, Zürich (1932).
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Pinl, M. Über die Gaußsche Krümmung der reellen Minimalflächen im R4 . Monatshefte für Mathematik 58, 27–32 (1954). https://doi.org/10.1007/BF01478560
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DOI: https://doi.org/10.1007/BF01478560