Normal acceleration and the second fundamental form

Try

For the graph z=x²+2y² at the origin, the upward unit normal is (0,0,1). A unit-speed curve initially pointing along the x-axis has normal acceleration two; one initially pointing along the y-axis has normal acceleration four. The surface bends differently in different directions.

The second fundamental form records that directional normal acceleration. We use the convention S=−dN for the shape operator and II(v,w)=⟨Sv,w⟩. Under this convention an outward-oriented sphere has negative principal curvatures. Gaussian curvature remains positive because it is their product.

Predict

Predict whether reversing N changes the first form. It does not: tangent lengths are unchanged. It reverses II and the shape operator because normal acceleration is measured against the opposite direction.

Predict whether two surface curves with the same unit tangent must have the same entire acceleration. Their normal components agree, but their tangential turning within the surface can differ.

Experiment

Compare the normal coefficients at the graph origin.

Unit tangent	Normal acceleration II(T,T)
(1,0,0)	2
(0,1,0)	4
(1/√2,1/√2,0)	3

The last entry follows from the quadratic form 2a²+4b², not from averaging vectors in space. The origin is special because the first form there is the identity; away from it, coordinate component length must be computed using g.

Understand

Let γ be a unit-speed curve on an oriented surface, with tangent T. Since T·N=0, differentiation gives γ″·N+T·dN(T)=0. Therefore γ″·N=⟨ST,T⟩=II(T,T). This proves the normal component depends only on the tangent direction and the surface at the point.

In a chart, differentiate N·X_j=0 in coordinate i. The result is N_i·X_j+N·X_ij=0. Hence the coefficients b_ij of II are b_ij=N·X_ij=−N_i·X_j. Equality of mixed partial derivatives makes this matrix symmetric. The coordinate formula uses second derivatives of the surface and the chosen unit normal.

For the radius-R sphere with outward N(p)=p/R, dN(v)=v/R. Thus S(v)=−v/R and II(v,w)=−⟨v,w⟩/R. A great-circle curve has inward acceleration, consistent with the negative dot product against the outward normal.

Apply

Repair “Positive curvature always means the surface bends upward.” The sign of each principal curvature depends on normal orientation, and Gaussian curvature compares the two signs rather than selecting an ambient up direction. A sphere can have both principal values negative while K is positive.

The normal acceleration formula also requires a unit tangent if it is being called normal curvature. For a curve with nonunit speed, the normal acceleration scales with speed squared. Tangential acceleration additionally contains changes of speed.

For optional depth, II can be defined for an immersed patch with a local normal even when no global orientation exists. Statements involving a global signed mean curvature require compatible normals; K remains well-defined under a normal sign change.

Continue

Write the normal orientation and S=−dN convention beside calculations to make signs reproducible. Continue to principal curvatures to diagonalize this quadratic bending information.