14.4. Normal Curvature and the Second Fun- damental Form
14.4. Normal Curvature and the Second Fun- damental Form
14.4. Normal Curvature and the Second Fun- damental Form
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Finally, we get<br />
<strong>and</strong><br />
A = u ′′<br />
1 + ∑ Γ 1 i j u ′ iu ′ j,<br />
i=1,2<br />
j=1,2<br />
B = u ′′<br />
2 + ∑ Γ 2 i j u ′ iu ′ j,<br />
i=1,2<br />
j=1,2<br />
<strong>Normal</strong> <strong>Curvature</strong> . . .<br />
Geodesic <strong>Curvature</strong> . . .<br />
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κ g<br />
−→ ng = ⎛<br />
⎜ ⎝ u ′′<br />
1 + ∑ i=1,2<br />
j=1,2<br />
⎞<br />
Γ 1 i j u ′ iu ′ ⎟<br />
j⎠ X u +<br />
⎛<br />
⎜<br />
⎝u ′′<br />
2 + ∑ i=1,2<br />
j=1,2<br />
⎞<br />
Γ 2 i j u ′ iu ′ ⎟<br />
j⎠ X v .<br />
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We summarize all <strong>the</strong> above in <strong>the</strong> following lemma.<br />
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