Affine surfaces in \(\mathbb R^4\) with planar geodesics with respect to the affine metric (Q2567420)
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| Language | Label | Description | Also known as |
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| English | Affine surfaces in \(\mathbb R^4\) with planar geodesics with respect to the affine metric |
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Affine surfaces in \(\mathbb R^4\) with planar geodesics with respect to the affine metric (English)
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4 October 2005
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The present paper characterizes affine surfaces in \(\mathbb R^4\) with planar geodesics with respect to the affine metric. The main theorem is: The only nondegenerate surfaces in \(\mathbb R^4\) whose geodesics with respect to the Levi-Civita connection of the affine metric are planar are the surfaces \(x(u,\nu)=(u,\nu,u\nu ,u^2-\nu^2)\) and \(x(u,\nu)=(u,\nu,u^2,\nu^2)\).
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affine connection
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metric connection
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geodesic
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