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The paper “Existence of universal connections” by Narasimhan, M. S.; Ramanan, S. proves that the Grassmanian is universal for connections not just bundles. That is any connection in a U(n) or O(n) bundle is pulled back from the canonical connection in the appropriate grassmanian by a map. Since the canonical connection has the desired form so does so does the original connection. They also estimate the required rank of the complement.

To clarify a bit. The tautological bundle over the Grassmannian $\gamma_k\to Gr_k(\mathbb{R}^N)$ has a complement $\gamma^\bot$ the bundle whose fiber at a subspace $V$ is the ortho-complement of $V$ in $\mathbb{R}^N$. It follows that $\gamma\oplus\gamma^\bot =Gr_k(\mathbb{R}^N) \times \mathbb{R}^N$. The connection in this paper is the connection induced from the trivial connection by projection.

The paper “Existence of universal connections” by Narasimhan, M. S.; Ramanan, S. proves that the Grassmannian is universal for connections not just bundles. That is any connection in a U(n) or O(n) bundle is pulled back from the canonical connection in the appropriate Grassmannian by a map. Since the canonical connection has the desired form so does so does the original connection. They also estimate the required rank of the complement.

To clarify a bit. The tautological bundle over the Grassmannian $\gamma_k\to \operatorname{Gr}_k(\mathbb{R}^N)$ has a complement $\gamma_k^\perp$ the bundle whose fiber at a subspace $V$ is the ortho-complement of $V$ in $\mathbb{R}^N$. It follows that $\gamma_k\oplus\gamma_k^\perp =\operatorname{Gr}_k(\mathbb{R}^N) \times \mathbb{R}^N$. The connection in this paper is the connection induced from the trivial connection by projection.

The paper “Existence of universal connections” by Narasimhan, M. S.; Ramanan, S. proves that the Grassmanian is universal for connections not just bundles. That is any connection in a U(n) or O(n) bundle is pulled back from the canonical connection in the appropriate grassmanian by a map. Since the canonical connection has the desired form so does so does the original connection. They also estimate the required rank of the complement.

The paper “Existence of universal connections” by Narasimhan, M. S.; Ramanan, S. proves that the Grassmanian is universal for connections not just bundles. That is any connection in a U(n) or O(n) bundle is pulled back from the canonical connection in the appropriate grassmanian by a map. Since the canonical connection has the desired form so does so does the original connection. They also estimate the required rank of the complement.

To clarify a bit. The tautological bundle over the Grassmannian $\gamma_k\to Gr_k(\mathbb{R}^N)$ has a complement $\gamma^\bot$ the bundle whose fiber at a subspace $V$ is the ortho-complement of $V$ in $\mathbb{R}^N$. It follows that $\gamma\oplus\gamma^\bot =Gr_k(\mathbb{R}^N) \times \mathbb{R}^N$. The connection in this paper is the connection induced from the trivial connection by projection.

The paper “Existence of universal connections” by Narasimhan, M. S.; Ramanan, S. proves that the Grassmanian is universal for connections not just bundles. That is any connection in a U(n) or O(n) bundle is pulled back from the canonical connection the appropriate grassmanian by a map. Since the canonical connection has the desired form so does so does the original connection. They also estimate the required rank of the complement.

The paper “Existence of universal connections” by Narasimhan, M. S.; Ramanan, S. proves that the Grassmanian is universal for connections not just bundles. That is any connection in a U(n) or O(n) bundle is pulled back from the canonical connection in the appropriate grassmanian by a map. Since the canonical connection has the desired form so does so does the original connection. They also estimate the required rank of the complement.