1
$\begingroup$

In Ana Cannas de silva's book Lectures on Symplectic geometry he defines a positive inner product to be smooth when for any vector field $v$ the function $x \mapsto g_x(v_x, v_x)$ is smooth.

Some other literature (i.e my lecture notes and wikipedia) require that for any two different vector fields $v,w$ we have that $x \mapsto g_x(v_x, w_x)$ is a smooth maps.

Are these two conditions equivalent? If so, how?

$\endgroup$
2
$\begingroup$

Hint: If $b(-,-)$ is a symmetric bilinear form in characteristic $\neq 2$, then $$b(x,y) = \frac{1}{2}\left(b(x+y,x+y) - b(x,x) - b(y,y)\right).$$

$\endgroup$

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.