Questions: The Second Fundamental Form, Gaussian Curvature, and Mean Curvature

Question 1

What are the Gaussian and mean curvatures of a circular cylinder of radius r (using the inward normal)?

• A. K = 1/r², H = 1/r
• B. K = 0, H = 1/(2r)
• C. K = 0, H = 0
• D. K = 1/r, H = 1/(2r)

Answer: (B)

Around the cylinder the normal curvature is 1/r, and along the rulings it is 0. So k₁ = 1/r and k₂ = 0, giving K = k₁k₂ = 0 and H = (1/r + 0)/2 = 1/(2r).

Question 2

Find the Gaussian curvature of the surface z = x² - y² at the origin.

Model answer: K = -4

At the origin the tangent plane is horizontal, so the second fundamental form is the Hessian of f(x, y) = x² - y², with f_xx = 2, f_yy = -2, f_xy = 0. The principal curvatures are 2 and -2, so K = -4. The negative sign marks a saddle (hyperbolic) point.

Question 3

Reversing the choice of unit normal changes the sign of the Gaussian curvature.

• T. True
• F. False

Answer: False

Replacing N by -N negates the shape operator, so both principal curvatures change sign. Their product K = k₁k₂ is unchanged, while the mean curvature H changes sign.

Question 4

A sphere of radius R has every normal curvature equal to 1/R (with the inward normal). What are K and H?

Model answer: K = 1/R², H = 1/R

Every direction is principal with curvature 1/R, so k₁ = k₂ = 1/R. Then K = 1/R² and H = 1/R. Points of a sphere are elliptic, and a sphere has constant positive Gaussian curvature.