14.4. Normal Curvature and the Second Fun- damental Form

However,
Normal Curvature . . .
κ −→ n = κ N N + κ g
−→ ng ,
Geodesic Curvature . . .
and since N is normal to the tangent space,
N · X u = N · X v = 0, and by dotting
κ g
−→ ng = AX u + BX v
with X u and X v , since E = X u · X u , F = X u · X v , and G =
X v · X v , we get the equations:
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κ −→ n · X u = EA + F B,
κ −→ n · X v = F A + GB.
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On the other hand,
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κ −→ n = X ′′ = X u u ′′ + X v v ′′ + X uu (u ′ ) 2 + 2X uv u ′ v ′ + X vv (v ′ ) 2 .
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