Differential Geometry Quiz

A surface in differential geometry is defined as:

• A one-dimensional object

• A two-dimensional manifold

• A three-dimensional volume

• A point in space

A curve on a surface is a curve that:

• Lies entirely on the surface

• Is always a straight line

• Exists only in Euclidean space

• Is always closed

A surface of revolution is obtained by rotating:

• A plane around a fixed point

• A curve around an axis

• A line around a circle

• A polygon around an edge

The helicoid is a surface generated by:

• Rotating a straight line while translating it along an axis

• Rotating a circle around another circle

• Twisting a parabola around an axis

• Translating a line in space without rotation

Metric direction coefficients are associated with:

• The curvature of a surface

• The distances measured along different directions on a surface

• The topology of a surface

• The area of a surface

Two surfaces are said to be isometric if:

• They have the same curvature at every point

• They can be transformed into one another without stretching

• They are mirror images of each other

• Their volume is equal

The Gaussian curvature of a surface is intrinsic because it depends on:

• The external embedding of the surface

• The way the surface is bent in space

• The intrinsic distances on the surface

• The normal vector field

Which of the following is an example of a surface of revolution?

• A plane

• A cylinder

• A torus

• A sphere

The term helicoid is derived from the Greek word meaning:

• Spiral

• Flat

• Bent

• Circle

A surface is said to be developable if:

• It has zero Gaussian curvature

• It is compact and closed

• It has constant mean curvature

• It is formed by revolving a straight line