Metric induced by a linear combinations of inner products on a manifold

Question

I have encountered a situation that assigns not just one inner product, but a linear combinations of inner products to each point of a manifold. Specifically, each point x in $X^n$ is assigned:

$$ \sum_{i=1}^m \lambda_i \langle \mathbf{v}, \mathbf{u} \rangle_i $$

for all vectors $\mathbf{v}$ and $\mathbf{u}$ at x.

What is the connection and metric that would parallel transport this structure?