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I have read different matrials on Calculus of Variations, but I still do not grasp the intuition of transversality condition. From textbooks, I can only roughly get an idea that with a transversality condition, I will get a unique solution. But I really would like to read some intuitions behind this idea. Please anyone provides some intuitions here.

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Transversality is a geometric concept which, locally, is a statement about linear subspaces. If you have two linear subspaces the sum of which is the ambient space, they are called transversal. This is usually applied to the tangent spaces at the intersection set of smooth submanifolds of some ambient manifold, e.g. Euclidean space or some Hilbert or Banach manifold. Saying that these meet transversally in some given point means that their tangent spaces are transversal in the sense I mentioned earlier. The relevance of this concepts stems from the implicit function theorem, which in this situation guarantees that the intersection set is a 'nice' set (a submanifold)locally. This does not imply that the intersection set is unique (a point), this is true only if the sum of the dimensions of the two submanifolds is the dimension of the ambient space (a statement which is a more complicated to formulate in case of infinite dimensions)

To understand the relevance of the concept it might help to look at the extreme opposite situation, which consists of two manifolds which, in some point, meet tangentially. In that case the intersection set of these manifolds may be extremely complicated. Think of the graphs of functions $f,g$ on $\mathbb{R}^2$ such that $f(0)=g(0)$ and $\nabla f(0) = \nabla g(0)$.

In general, thinking of smooth submanifolds of $\mathbb{R}^3$ and their intersections is a good starting point to get some intuitive feeling on transversality.

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would you please recommend some book or notes to read? –  Flying pig Jul 11 '12 at 7:06
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@Flyingpig Any textbook on differential topology should do, you may try Hirsch's monograph. It has a dedicated chapter on transversality. –  user20266 Jul 11 '12 at 17:03

Simple intuition of transversality is manifolds untersecting under nonzero angle. As result their tangent spaces at intersection spawn ambient space (tangent space of ambient manifold)

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