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Let $M$ be a $n$-dimensional smooth riemannian manifold and $\varphi\colon(-\varepsilon,\varepsilon)\rightarrow M$ an embedding. $\varphi$ will denote the image of $\varphi$, too.

Consider the bundle of orthonormal $(n-k)$-frames in $M$ over $\varphi$, where $k>2$, which are orthogonal to $\varphi$. The fiber of this bundle is the orthonormal stiefel manifold $V_{n-k}(\mathbb{R}^{n-1})$, which is connected, since $n-k<n-1$.

My first idea to show, that this bundle is smoothly trivial was, to transport one fixed fiber via parallel transport along $\varphi$ of the Levi-Civita-connection of $M$. This works, because parallel transport along $\varphi$ gives me isometries, which leave vectors tangent to $\varphi$ fixed. (Just to be sure, this works, right?)

Now take a global section $\eta$ of this bundle and consider the bundle of orthonormal $(k-1)$-frames in $M$ over $\varphi$, which are orthogonal to $\varphi$ and to $\eta$. This gives me a bundle with fiber $V_{k-1}(\mathbb{R}^{n-1-(n-k)})=V_{k-1}(\mathbb{R}^{k+1})$.

I am searching for an argument to show, that this second bundle is also smoothly trivial. The proof considered before doesn't work, because parallel transport along $\varphi$ doesn't preserve beeing orthogonal to $\eta$ in general. Can someone help me out by giving me a hint how to prove that?

I don't want to use any classification results or any sophisticated bundle-theory. I think there must be an more-or-less elementary argument like the one of my first idea.

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1 Answer 1

The first idea only works when $\phi$ is a reparametrized geodesic. Write $\dot \phi= \tfrac{d}{dt}\phi$ and denote parallel transport along $\phi$ by $P_{\phi}$. Then $P_{\phi}(\dot\phi(0))$ remains tangent to $\phi$ iff there is an $\alpha:(-\varepsilon,\varepsilon)\to \mathbb{R}$ s.t. $P_{\phi}(\dot \phi(0))(t)=\alpha(t)\dot \phi(t)$ and then $\nabla_{t}(\alpha\dot \phi)=0$, i.e. $\nabla_t(\tfrac{d}{dt}\phi(\int_{0}^{t}\alpha(\tau)d\tau))=0$.

If $\phi$ is not a reparametrized geodesic then assume it is normalised, choose an orthonormal (n-k)-frame $\{e_{i}\}$ at $T_{\phi(0)}M$ which is orthogonal to $\dot\phi(0)$ and solve the system of equations: $$\nabla_{t}E_{i}=\langle E_{i},\nabla_{t}\dot \phi\rangle \dot \phi,\quad E_{i}(0)=e_{i}$$ The solutions satisfy $\langle \nabla_{t}E_{i},\dot \phi\rangle=\langle E_{i},\nabla_{t}\dot \phi\rangle$ and therefore $\tfrac{d}{dt}\langle E_{i},\dot \phi\rangle=0$ and the orthogonality at $t=0$ is preserved for all $t$. Then also $\langle \nabla_{t}E_{i},E_{j}\rangle=0$ and therefore orthonormality of the $\{e_{i}\}$ is preserved too.

Something similar will work for you second question, but I think you will need to involve $\eta$ in the system of equations.

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