Let  $(M,g)$  be  a Riemannian  manifold  which  admit a non vanishing  vector  field.(That is $\chi(M)=0$  when $M$ is  a compact  manifold). We  pull back The  symplectic  structure of  the  cotangent  bundle  to the  $2$-form  $\omega$ on $TM$. 

>Is there necessarily  a  non vanishing  vector  field $X$ on $M$ for  which the  following submanifold of $(TM, \omega)$  would  be a  symplectic  submanifold?

>$$\{v_p\in TM \mid |v_p|=1, v \perp X(p)\}$$

>where $v_p$ is  a  vector in $TM$ based at point $p\in M$.

The  motivation for  this  question is the  following:

We  would  like  to  find  some  symplectic  submanifolds  of  $TM$ which  are in the  form of  a  sub vector bundle  of  the tangent  bundle  or  sub fiber bundle  of  unite tangent  bundle.

In the  standard coordinate $(x_1,x_2,\ldots,x_n,y_1,y_2,\ldots,y_n)$, the  elementary  examples  of  symplectic  submanifolds are $$(x_1,x_2,\ldots,x_k,0,0,\ldots,0,y_1,y_2,\ldots,y_k,0,0\ldots,0)$$

In  such  elementary example  we loose  the  whole  base  space.