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When reading about Pesin theory I've run into the necessity of defining a metric on the Grassmannian bundle of a compact Riemannian manifold $M$. More specifically a fiber at $x \in M$ in the Grassmannian bundle consists of the space of $k$-dimensional subspaces of the tangent space $T_x M$. In the notes I'm reading, they say that the Riemannian metric "naturally induces" distances in this bundle, but I'm not sure what is this natural way.

I have defined a metric in the Grassmannian by saying that the distance $d(A,B)$ between two subspaces $A$ and $B$ is the norm of the linear operator $P_A - P_B$ where $P_X$ is the projection on the subspace $X$.

One guess I have for this naturally induced metric is that if $A_x \subset T_xM$ is a subspace at $x \in M$ and $B_y \subset T_yM$ is a subspace at $y \in M$, then their distance could be taken to be $$\rho(x,y) + d(A_x,h(B_y)),$$ where $h \colon T_y M \to T_x M$ is a parallel transport defined by a geodesic between $x$ and $y$ and $\rho$ is the distance given by the Riemannian metric. I'm not sure if this makes even sense, though.

In any case the resulting metric should be such that when we use Whitney embedding theorem, we get that in the embedded submanifold $M' \subset \mathbb{R}^N$ the distance given by the original Riemannian metric is equivalent to the Euclidean distance of $\mathbb{R}^N$ and the distance in the Grassmannian bundle then is simply equivalent to $$\|x - y\| + d(A_x,A_y)$$ where $A_x$ and $A_y$ are now considered to be subspaces of (the same space) $\mathbb{R}^N$.

So my questions are:

  1. Is there a natural way of defining metrics or even Riemannian metrics in a fiber bundle?
  2. If the answer to 1 is negative, is there some popular way of "naturally inducing" distances in the Grassmannian bundle?
  3. Finally, if no better answer is available, what do you think about my guess?

References to literature would also be highly appreciated.

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You may want to consult Section 2 here and also this article for a construction of a natural metric. –  t.b. Oct 26 '11 at 11:43
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