Umbilics of Surfaces in the Lorentz–Minkowski 3-Space

<jats:title>Abstract</jats:title><jats:p>In this paper, we prove several fundamental properties on umbilics of a space-like or time-like surface in the Lorentz–Minkowski space<jats:inline-formula><jats:alternatives><jats:tex-math>$${{\mathbb {L}}}^3$$</jats:tex-math><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:msup><mml:mrow><mml:mi>L</mml:mi></mml:mrow><mml:mn>3</mml:mn></mml:msup></mml:math></jats:alternatives></jats:inline-formula>. In particular, we show that the local behavior of the curvature line flows of the germ of a space-like surface in<jats:inline-formula><jats:alternatives><jats:tex-math>$${{\mathbb {L}}}^3$$</jats:tex-math><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:msup><mml:mrow><mml:mi>L</mml:mi></mml:mrow><mml:mn>3</mml:mn></mml:msup></mml:math></jats:alternatives></jats:inline-formula>is essentially the same as that of a surface in Euclidean space. As a consequence, for each positive integer<jats:italic>m</jats:italic>, there exists a germ of a space-like surface with an isolated<jats:inline-formula><jats:alternatives><jats:tex-math>$$C^{\infty }$$</jats:tex-math><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:msup><mml:mi>C</mml:mi><mml:mi>∞</mml:mi></mml:msup></mml:math></jats:alternatives></jats:inline-formula>-umbilic (resp. <jats:inline-formula><jats:alternatives><jats:tex-math>$$C^1$$</jats:tex-math><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:msup><mml:mi>C</mml:mi><mml:mn>1</mml:mn></mml:msup></mml:math></jats:alternatives></jats:inline-formula>-umbilic) of index<jats:inline-formula><jats:alternatives><jats:tex-math>$$(3-m)/2$$</jats:tex-math><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mrow><mml:mo>(</mml:mo><mml:mn>3</mml:mn><mml:mo>-</mml:mo><mml:mi>m</mml:mi><mml:mo>)</mml:mo><mml:mo>/</mml:mo><mml:mn>2</mml:mn></mml:mrow></mml:math></jats:alternatives></jats:inline-formula>(resp.<jats:inline-formula><jats:alternatives><jats:tex-math>$$1+m/2$$</jats:tex-math><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mrow><mml:mn>1</mml:mn><mml:mo>+</mml:mo><mml:mi>m</mml:mi><mml:mo>/</mml:mo><mml:mn>2</mml:mn></mml:mrow></mml:math></jats:alternatives></jats:inline-formula>). We also show that the indices of isolated umbilics of time-like surfaces in<jats:inline-formula><jats:alternatives><jats:tex-math>$${{\mathbb {L}}}^3$$</jats:tex-math><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:msup><mml:mrow><mml:mi>L</mml:mi></mml:mrow><mml:mn>3</mml:mn></mml:msup></mml:math></jats:alternatives></jats:inline-formula>that are not the accumulation points of quasi-umbilics are always equal to zero. On the other hand, when quasi-umbilics accumulate, there exist countably many germs of time-like surfaces which admit an isolated umbilic with non-zero indices.</jats:p>