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The unit normal to a two dimensional surface in 3 space maps the surface into the unit sphere. The induced vector bundle under this map, the Gauss map,is the surface's tangent bundle as can be seen from parallel translating its tangent planes to the origin.

The induced connection on the surface's tangent bundle - i.e. the pull back under the Gauss map of the standard connection on the sphere - defines a connection on the surface.

Here are some questions about the induced connection.

  • Given a second embedding of the surface, the Gauss map induces a second connection. This second Gauss map is homotopic to the first - since both classify the surface's tangent bundle. One can imagine deforming one surface into the other to obtain a homotopy and simultaneously a homotopy between the two connections. Does such a deformation always exist? If not can the homotopy be realized as a 1 parameter family of embeddings in a higher dimensional Euclidean space?

  • Is the induced connection just the Levi-Civita connection on the surface that it inherits from its embedding in 3 space?

If so this seems quite strange since the derivative of the Gauss map is not usually the induced map of vector bundles.

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The space of connections on a manifold is an affine space, literally you can take $(1-t)$ times your first connection plus $t$ times your second connection, this works at the level of Christoffel symbols, and also at the level of Ehresmann connections. That answers your first question.

For your second question, yes there's a direct formula for the submanifold Levi-Cevita connection in terms of the Gauss map. It's a little complicated but there is such a formula.

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  • $\begingroup$ Ok I will try to derive the formula for the Levi_Cevita connection in terms of the Gauss map. For me this is a revealing example since the differential of the Gauss map is not the bundle map from the surface to the sphere. I understand that any two connections are homotopic in the space of connections. what I was asking is whether this homotopy can be realized geometrically as a smooth deformation of the two embeddings of the surface - in 3 space or if not in 3 space in some higher dimensional Euclidean space. $\endgroup$ – lavinia May 29 '13 at 15:21
  • $\begingroup$ Generally no such deformation exits. For example, you can embed $S^1 \times S^1$ in a way that makes it knotted. But for $S^2$, there is only one isotopy class of embedding, this is known as the Alexander Theorem, so you can make such a deformation. But that's a very special case. $\endgroup$ – Ryan Budney May 29 '13 at 16:32

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