Talk:Second fundamental form

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Something here is wrong:

\langle R(u,v)w,z\rangle =\langle \mathrm I\!\mathrm I(u,z),\mathrm I\!\mathrm I(v,w)\rangle-\langle \mathrm I\!\mathrm I(u,w),\mathrm I\!\mathrm I(v,z)\rangle.
\langle R_N(u,v)w,z\rangle = \langle R_M(u,v)w,z\rangle+\langle \mathrm I\!\mathrm I(u,z),\mathrm I\!\mathrm I(v,w)\rangle-\langle \mathrm I\!\mathrm I(u,w),\mathrm I\!\mathrm I(v,z)\rangle.

The second fundamental form is a bilinear operator. II(u,v) is a number, not a vector. Why are we taking inner products of numbers?

68.1.154.81 (talk) 21:11, 13 March 2008 (UTC)

Thanks for catching that. At this point in the text, the second fundamental form refers to a normal-bundle valued bilinear form (i.e., it's vector valued, not scalar valued). I have added a section break to set up the contrast a little better, but it probably needs still more clarification. silly rabbit (talk) 21:33, 13 March 2008 (UTC)