Differential geometry

• What Is the Purpose of a Cusp?
  A cusp marks a singular point where a curve reverses direction with a common limiting tangent, useful in geometry and applications.
• Is the Pushforward the Same as the Differential?
  For a smooth map, its differential at each point is the pushforward on tangent vectors; scalar-valued functions give covectors, not tangent vectors.
• Christoffel Symbols in Terms of $\sqrt{|g|}$
  The contracted Christoffel symbols satisfy Γᵘᵢᵤ = ∂ᵢ ln√|g| = (1/√|g|)∂ᵢ√|g|, with g the metric determinant.
• How Is the Second Covariant Derivative of a Scalar Computed?
  Compute a scalar's second covariant derivative by differentiating its gradient as a covector and subtracting the Christoffel-symbol term.
• What Is the Coordinate-Free Definition of the Nabla Operator?
  Nabla is not one universal coordinate-free vector: the gradient comes from a metric, while covariant differentiation comes from a connection.
• How to Calculate the Riemann Tensor of a 3-Sphere
  For the round 3-sphere of radius a, the Riemann tensor has constant sectional curvature 1/a²; here is its coordinate form and derivation.