Differential Geometry

Curves and Surfaces, Curvature

• 79The shape operator: real eigenvalues, but not always symmetric as a raw matrixThe shape operator: real eigenvalues, but not always symmetric as a raw matrixMSET September 2021Curves and Surfaces, CurvatureHard
• 80A Frenet-frame combination that collapses to zeroA Frenet-frame combination that collapses to zeroMSET September 2021Curves and Surfaces, CurvatureMedium
• 81Cylinders are developable — Gaussian curvature vanishes identicallyCylinders are developable — Gaussian curvature vanishes identicallyMSET September 2021Curves and Surfaces, CurvatureEasy
• 80A unit first-fundamental-form determinant means area is preserved, not necessarily anglesA unit first-fundamental-form determinant means area is preserved, not necessarily anglesMSET December 2020Curves and Surfaces, CurvatureMedium
• 81A hyperbolic paraboloid has strictly negative Gaussian curvature everywhereA hyperbolic paraboloid has strictly negative Gaussian curvature everywhereMSET June 2019Curves and Surfaces, CurvatureEasy
• 56The curve is not unit-speed, so alpha"=2 is not its curvatureThe curve is not unit-speed, so $|\alpha "|=2$ is not its curvatureMSET January 2018 Paper IIICurves and Surfaces, CurvatureMedium
• 57A curve lying in the plane x+y=0 has zero torsionA curve lying in the plane $x+y=0$ has zero torsionMSET January 2018 Paper IIICurves and Surfaces, CurvatureMedium
• 58The unit cylinder is locally isometric to the plane: E=G=1, F=0The unit cylinder is locally isometric to the plane: $E=G=1$, $F=0$MSET January 2018 Paper IIICurves and Surfaces, CurvatureEasy
• 56Frenet–Serret with kappa=2, tau=s+1 (convention b'=tau n)Frenet–Serret with $\kappa =2$, $\tau =s+1$ (convention $b^{\prime }=\tau n$)MSET April 2017 Paper IIICurves and Surfaces, CurvatureMedium
• 57Twisted cubic at the origin: alpha'timesalpha"/alpha'^3=2Twisted cubic at the origin: $|{\alpha }^{\prime }\times \alpha "|/|{\alpha }^{\prime }{|}^3=2$MSET April 2017 Paper IIICurves and Surfaces, CurvatureEasy
• 58Helicoid: E=v^2+a^2, F=0, G=1Helicoid: $E=v^2+a^2$, $F=0$, $G=1$MSET April 2017 Paper IIICurves and Surfaces, CurvatureEasy