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 -

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- April 29th 2010, 01:35 PMAlexanderWDifferentiable submanifold
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 - - April 29th 2010, 02:08 PMLaurent
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 .

- April 29th 2010, 02:57 PMAlexanderW
- April 30th 2010, 04:09 AMLaurent