Differential Geometry

Differential geometry uses the techniques of differential and integral calculus to study the geometry of curves and surfaces.

Geometry of Curves

A space curve is defined by a vector function $\mathbf{r}(t)$ . The key invariants are curvature ( $\kappa$ ) and torsion ( $\tau$ ).

Interactive Lab

import numpy as np

def calculate_curvature(t_val, dr_dt, d2r_dt2):
  v1 = dr_dt(t_val)
  v2 = d2r_dt2(t_val)
  cross_product = np.cross(v1, v2)
  norm_v1 = np.linalg.norm(v1)
  kappa = np.linalg.norm(cross_product) / (norm_v1**3)
  return kappa

# Helix: r(t) = (cos t, sin t, t)
dr_dt = lambda t: np.array([-np.sin(t), np.cos(t), 1])
d2r_dt2 = lambda t: np.array([-np.cos(t), -np.sin(t), 0])

print(f"Curvature of Helix at t=0: {calculate_curvature(0, dr_dt, d2r_dt2):.3f}")
print("Curvature is constant for a helix.")

python

1import numpy as np

3def calculate_curvature(t_val, dr_dt, d2r_dt2):

4 v1 = dr_dt(t_val)

5 v2 = d2r_dt2(t_val)

6 cross_product = np.cross(v1, v2)

7 norm_v1 = np.linalg.norm(v1)

8 kappa = np.linalg.norm(cross_product) / (norm_v1**3)

9 return kappa

11# Helix: r(t) = (cos t, sin t, t)

12dr_dt = lambda t: np.array([-np.sin(t), np.cos(t), 1])

13d2r_dt2 = lambda t: np.array([-np.cos(t), -np.sin(t), 0])

15print(f"Curvature of Helix at t=0: {calculate_curvature(0, dr_dt, d2r_dt2):.3f}")

16print("Curvature is constant for a helix.")

Gaussian Curvature

For surfaces, Gaussian Curvature ( $K$ ) is the product of the two principal curvatures. It is an intrinsic property, meaning it can be determined by measuring distances along the surface without reference to the surrounding space.

If we bend a flat sheet of paper into a cylinder, does its Gaussian curvature change?

Gauss-Bonnet Theorem

The Gauss-Bonnet Theorem links the total curvature of a surface to its topology (Euler characteristic $\chi$ ): $\int_M K dA + \int_{\partial M} k_g ds = 2\pi \chi(M)$

What is the Euler characteristic of a sphere?

Interactive Lab

def euler_characteristic(v, e, f):
  return v - e + f

# Cube: 8 vertices, 12 edges, 6 faces
print(f"Cube χ: {euler_characteristic(8, 12, 6)}")
# Tetrahedron: 4 vertices, 6 edges, 4 faces
print(f"Tetrahedron χ: {euler_characteristic(4, 6, 4)}")

python

1def euler_characteristic(v, e, f):

2 return v - e + f

4# Cube: 8 vertices, 12 edges, 6 faces

5print(f"Cube χ: {euler_characteristic(8, 12, 6)}")

6# Tetrahedron: 4 vertices, 6 edges, 4 faces

7print(f"Tetrahedron χ: {euler_characteristic(4, 6, 4)}")