Suppose Riemanian manifold $(M,\alpha)$ and 1-form $\beta$ such that
$$\nabla\beta(X,Y)=|| \beta^\sharp||^2 \alpha(X,Y)-\beta(X)\beta(Y)$$
and Questions are :
$1.$ $\beta $ is closed and it has local form $\beta=df$
$2.$ $\beta^\sharp$ has a constant lenght.
$3.$ Integral curve of $\beta^\sharp$
$3.$ $$R(X,Y)\beta ^\sharp=K^2(\beta(X)Y-\beta(Y)X)$$ and sectional curvature is $-K^2$ for any $2-$plane containing the vector field $\beta^\sharp $
My attempt to answers Above: look locally at $\beta$ first let me write .
$$\bbox[yellow]{\beta=\beta_idx^i\to \beta^\sharp =\beta_i\alpha^{ij}\partial_j} $$
$$\bbox[pink]{||\beta^\sharp||^2=\alpha(\beta_i\alpha^{ij}\partial_j,\beta_k\alpha^{kl}\partial_l)=\beta_l\beta_l\alpha^{kl}}$$
$$\bbox[lightgreen]{\nabla \beta(X,Y)=\nabla_X\beta(Y)=\beta(\nabla_X^Y)}$$
If take $X=\partial i,Y=\partial j$ then we have $$\bbox[orange]{\nabla \beta(\partial i,\partial j)=\partial i\beta_j-\Gamma_{ij}^k\beta_k=||\beta^\sharp||^2a\alpha_{ij}-\beta_i\beta_j}$$
to answer first one
$d\overbrace{\beta(\partial ij, \partial )=\partial i\beta(\partial j)-\partial j\beta(\partial i)-\beta([\partial i,\partial j])}^{\text{ $\Gamma_{ij}^k=\Gamma_{ji}^k$}}$=0
Just a hint will be appreciate for each part.