Hello.
Let be a differentiable submanifold (with the induced metrics from ), ,
and has a minimum .
Proof, that the vector is orthogonal to .
is the tangent space at .
Thanks in advance for help!
Buy,
- Alexander -
For any differentiable curve on such that , the function has a minimum at 0, hence its derivative at this point is 0, which gives , i.e. : is orthogonal to . Since is spanned by the vectors of the form (this is even one possible definition), we deduce that is orthogonal to .