| Summary: | We classify Weingarten conoids in the real special linear group <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>SL</mi><mo>(</mo><mn>2</mn><mo>,</mo><mi mathvariant="double-struck">R</mi><mo>)</mo></mrow></semantics></math></inline-formula>. In particular, there is no linear Weingarten nontrivial conoids in <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>SL</mi><mo>(</mo><mn>2</mn><mo>,</mo><mi mathvariant="double-struck">R</mi><mo>)</mo></mrow></semantics></math></inline-formula>. We also prove that the only conoids in <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>SL</mi><mo>(</mo><mn>2</mn><mo>,</mo><mi mathvariant="double-struck">R</mi><mo>)</mo></mrow></semantics></math></inline-formula> with constant Gaussian curvature are the flat ones. Finally, we show that any surface that is invariant under left translations of the subgroup <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mi mathvariant="script">N</mi></semantics></math></inline-formula> is a Weingarten surface.
|
|---|