I have the following setting: Let (M,g) be a Riemannian manifold and $\iota: M \to R^N$ some isometric embedding. This means especially that the connection $\nabla^{TM}$ of M is given by the ordinary Euklidean connection $\nabla$ on R^N composed with the projection on the subbundle TM of TR^N. I also have the normal bundle NM of M and can equip it with the via $\iota$ pulled back metric and connection.

Now I want to bound the Christoffel symbols (and its derivatives) of $\nabla^{NM}$ by the Christoffel symbols of $\nabla^{TM}$ and also bound the entries of the projection matrices $p_{TM}: R^N \to TM$ and $p_{NM}: R^N \to NM$. And vice versa.

For the computation of $\Gamma^{TM}$ I use Riemannian normal coordinates $\partial_{x_i}$. For the computation of $\Gamma^{NM}$ I take an orthonormal basis ${n_i}$ of NM at a point p and then parallel translate it along the radial geodesics of the Riemannian normal coordinate system. The entries of the projections I want to compute with respect to the standard coordinates $e_i$ of R^N.

I have the following: Let $g_{ij}$ and $h_{ij}$ be the local expressions for the metrics g and h (w.r.t. the frame $\partial_{x_i}$, resp. $n_i$) and let $g^{ij}$ and $h^{ij}$ be the inverse matrices. Then the projections are given by $p_{TM}(v) = g^{kl} \langle v, \partial_{x_l} \rangle \partial_{x_k}$ and analogously for $p_{NM}$. Then the Christoffel symbols are given by $\Gamma_{ij}^{k, TM} = g^{kl} \langle \nabla_{\partial_{x_i}} \partial_{x_j}, \partial_{x_l}\rangle$ and analogously for $\Gamma_{ij}^{k, NM}$ and the entries of the projection metrices are $(p_{TM})_{ij} = g^{kl} \langle e_j, \partial_{x_l}\rangle \langle e_i, \partial_{x_k}\rangle$ and analogously for $(p_{NM})_{ij}$.

Now if I suppose that all the $\Gamma_{ij}^{k, TM}$ are bounded as are all their derivatives, I do not see how to show the boundedness of the $(p_{TM})_{ij}$ and their derivatives. And even if somehow could show this, I don't see how to show then the bundedness of the $\Gamma_{ij}^{k, NM}$ and their derivatives.

Motivation for the whole question: A Riemannian manifold has bounded geometry, if the metric is complete, the injectivity radius positive and the curvature tensor and its covariant derivatives are uniformly bounded. One can show that this is equivalent to the statement, that the Christoffel symbols and its derivates are all uniformly bounded when computed in Riemannian normal coordinates (where the radii of all the coordinate balls are fixed). A bundle over a Riemannian manifold equipped with a connection has bounded geometry, if its curvature tensor and all its covariant derivatives are bounded. Here one can also show that this is equivalent to the statement, that the Christoffel symbols and its derivatives are uniformly bounded, when computed in such a frame as above (parallel translation of an orthonormal basis at a point along radial geodesics). Now I want to show, that if a Riemannian manifold has bounded geometry, its normal bundle (w.r.t. some isometric embedding into some R^N) has also bounded geometry. And I am also interested in a formulation of this via the entries of the projection matrices.

Thanks, Alexander Engel


closed as too localized by Willie Wong Mar 4 '11 at 16:36

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  • $\begingroup$ Maybe there is another way to solve the problem (which I mentioned in the motivation)? I would be happy with any solution. Alex $\endgroup$ – AlexE Feb 18 '11 at 10:03
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    $\begingroup$ Since I got no answer I posted the question at MathOverflow: mathoverflow.net/questions/57109. $\endgroup$ – AlexE Mar 2 '11 at 12:05
  • $\begingroup$ @Alex: since Bill Thurston gave what probably is the right answer (I still have to think it through a bit), would you be okay if I closed this question as "no longer relevant"? $\endgroup$ – Willie Wong Mar 3 '11 at 9:39
  • $\begingroup$ Yes, this question here can be closed. $\endgroup$ – AlexE Mar 4 '11 at 12:44
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    $\begingroup$ @Willie: Why don't we just post a CW answer with a link to the MO answer? $\endgroup$ – Aryabhata Mar 4 '11 at 23:52