Geometry of Surfaces Quiz

A surface where all points are umbilics must have:

• Constant mean curvature

• Zero Gaussian curvature

• Zero second fundamental form

• Constant principal curvatures

Which of the following is an example of a compact surface whose points are umbilics?

• Sphere

• Hyperboloid

• Helicoid

• Cylinder

Hilbert’s Lemma states that a compact surface with constant positive curvature must be:

• A plane

• A part of a sphere

• A developable surface

• A ruled surface

The Gaussian curvature of a compact surface of constant curvature is:

• Constant at every point

• Zero at some points

• Variable along geodesics

• Only dependent on principal curvatures

Which of the following is a compact surface of constant curvature?

• Torus

• Sphere

• Paraboloid

• Helicoid

A surface is complete if:

• All geodesics can be extended indefinitely

• It is compact without a boundary

• It has zero mean curvature

• It is a ruled surface

Which of the following surfaces is complete?

• A Euclidean plane

• A paraboloid

• A helicoid

• A truncated sphere

Hilbert’s Theorem states that:

• There is no complete surface of constant negative curvature in R3

• Minimal surfaces exist in every compact space

• Every compact surface has at least one umbilic point

• Geodesics on a sphere are straight lines

The implication of Hilbert’s Theorem is that:

• Hyperbolic surfaces cannot be isometrically embedded in R3

• The Gaussian curvature of a compact surface must be positive

• Geodesics always have constant curvature

• Umbilic points exist on all surfaces

Two points on a geodesic are conjugate if:

• The Jacobi field along the geodesic vanishes at both points

• They lie on the same principal curvature line

• Their geodesic curvature is equal

• They are umbilic points