| Summary: | In the 3-dimensional Euclidean space <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msup><mi>E</mi><mn>3</mn></msup></semantics></math></inline-formula>, a quadric surface is either ruled or of one of the following two kinds <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msup><mi>z</mi><mn>2</mn></msup><mo>=</mo><mi>a</mi><msup><mi>s</mi><mn>2</mn></msup><mo>+</mo><mi>b</mi><msup><mi>t</mi><mn>2</mn></msup><mo>+</mo><mi>c</mi><mo>,</mo><mi>a</mi><mi>b</mi><mi>c</mi><mo>≠</mo><mn>0</mn></mrow></semantics></math></inline-formula> or <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>z</mi><mo>=</mo><mfrac><mi>a</mi><mn>2</mn></mfrac><msup><mi>s</mi><mn>2</mn></msup><mo>+</mo><mfrac><mi>b</mi><mn>2</mn></mfrac><msup><mi>t</mi><mn>2</mn></msup><mo>,</mo><mi>a</mi><mo>></mo><mn>0</mn><mo>,</mo><mspace width="4pt"></mspace><mi>b</mi><mo>></mo><mn>0</mn></mrow></semantics></math></inline-formula>. In the present paper, we investigate these three kinds of surfaces whose Gauss map <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mi mathvariant="bold-italic">N</mi></semantics></math></inline-formula> satisfies the property <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msup><mo>Δ</mo><mrow><mi>I</mi><mi>I</mi></mrow></msup><mi mathvariant="bold-italic">N</mi><mo>=</mo><mi>Λ</mi><mi mathvariant="bold-italic">N</mi></mrow></semantics></math></inline-formula>, where <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mi>Λ</mi></semantics></math></inline-formula> is a square symmetric matrix of order 3, and <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msup><mo>Δ</mo><mrow><mi>I</mi><mi>I</mi></mrow></msup></semantics></math></inline-formula> denotes the Laplace operator of the second fundamental form <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>I</mi><mi>I</mi></mrow></semantics></math></inline-formula> of the surface. We prove that spheres with the nonzero symmetric matrix <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mi>Λ</mi></semantics></math></inline-formula>, and helicoids with <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mi>Λ</mi></semantics></math></inline-formula> as the corresponding zero matrix, are the only classes of surfaces satisfying the above given property.
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