I agree with the answer given by Peter Dalakov, and with the comments made so far. On the other hand, there is a solution as close as possible to the requirement.

We consider a default Riemannian metric $g_0$ on the vector bundle $E$. This is the special metric mentioned by Johannes NordstrÃ¶m in his comment. Any other metric can be obtained from this one, by applying a section from the bundle $GL(E)$ (having as fiber at $p\in M$ the general linear group of $E_p$). Any section of $GL(E)$, when applied to $g_0$, will give another Riemannian metric. Also, any section of $GL(E)$ can be written as $\exp(s)$, where $s$ is a section of the vector bundle $SL(E)$.

Hence, we can take as the desired vector bundle the bundle $SL(E)$. Any section $s$ of it will provide a transformation $\exp(s)$ which, when applied to the specially chosen metric $g_0$, gives another metric. Any metric can be obtained this way.

Note 1:

This works similarly for the Hermitian case.

Note 2:

If for the particular problem is necessary to work with generic linear combinations of the metrics themselves, then the solution proposed here will not be of use. Instead, one needs geometric methods which apply to metrics having variable signature, hence also being degenerate. Such methods were developed in arXiv:1105.0201, arXiv:1105.3404, and arXiv:1111.0646. One can define for example covariant derivatives for special tensor fields and differential forms, and also define the Riemann tensor $R_{abcd}$, although these constructions are apparently forbidden by $g_{ab}$ not being always invertible.

1

I agree with the answer given by Peter Dalakov, and with the comments made so far. On the other hand, there is a solution as close as possible to the requirement.

We consider a default Riemannian metric $g_0$ on the vector bundle $E$. This is the special metric mentioned by Johannes NordstrÃ¶m in his comment. Any other metric can be obtained from this one, by applying a section from the bundle $GL(E)$ (having as fiber at $p\in M$ the general linear group of $E_p$). Any section of $GL(E)$, when applied to $g_0$, will give another Riemannian metric. Also, any section of $GL(E)$ can be written as $\exp(s)$, where $s$ is a section of the vector bundle $SL(E)$.

Hence, we can take as the desired vector bundle the bundle $SL(E)$. Any section $s$ of it will provide a transformation $\exp(s)$ which, when applied to the specially chosen metric $g_0$, gives another metric. Any metric can be obtained this way.