Abstract
In this chapter we recall the notion of stable map between two manifolds.1 It is convenient to introduce the terminology in arbitrary dimension, and in a rather abstract setting.
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Notes
- 1.
- 2.
The map \(\mathcal{F}\) keeps the orientation: for instance, when n = 2, a positively oriented Jordan curve in \(\mathbb{R}^{2}\) is mapped through \(\mathcal{F}\) into a positively oriented Jordan curve.
- 3.
In the terminology of Thom, Morse functions are called correct, and Morse functions with distinct critical values are called excellent [6].
- 4.
Not quite, since \(\mathbb{R}\) is not closed. However we require functions to behave “nicely” outside some interval [a, b], e.g. by forcing them to have constant nonzero derivative.
- 5.
In this book a knot is a \(\mathcal{C}^{\infty }\) embedding of \(\mathbb{S}^{1}\) in \(\mathbb{R}^{3}\), hence in particular a tame knot in the usual terminology; see, for instance, [8, p. 5].
- 6.
Therefore, each of these curves is the embedded image of \(\mathbb{S}^{1}\) into \(\mathcal{X}\).
- 7.
We warn the reader that these cusps, belonging to the source manifold \(\mathcal{X}\), should not be confused with the cusps of an apparent contour, which lie in the target manifold \(\mathcal{Y}\).
- 8.
Sometimes called the cone on a figure-eight curve.
- 9.
See, for instance, [10, Chapter II, Proposition 5.8].
- 10.
- 11.
M is an abstract manifold, not necessarily oriented or connected. We shall be mostly interested (for instance, in Sect. 3.2) in the case when M can be embedded in \(\mathbb{R}^{3}\), which gives, in particular, an orientation to \(M\).
- 12.
Locally, each cusp of the apparent contour is diffeomorphic to the simple (or ordinary, see, for instance, [3, p. 115]) cusp, which has the form \(\{(x_{1},x_{2}): x_{2}^{2} = x_{1}^{3}\}\) or equivalently, in a parametric form, \((t^{2},t^{3})\) for a real parameter t in a neighbourhood of the origin.
- 13.
The component is, sometimes, called “irreducible” (with cusps and double points); see, e.g., [22].
- 14.
- 15.
- 16.
- 17.
The first property means that H admits an extension of class \(\mathcal{C}^{\infty }\) on an open set of \(\mathbb{R}^{n+1}\) containing \(\mathbb{R}^{n} \times [0, 1]\).
- 18.
Consistently, we set h t (⋅ ) = h(⋅ , t).
- 19.
We recall that, if V is a \(\mathcal{C}^{\infty }\) orientable n-dimensional manifold without boundary, and if Diff+(V ) denotes the group of positive diffeomorphisms of V endowed with the \(\mathcal{C}^{\infty }(V,V )\) topology, then the orbits coincide with the connected components; see [7, p. 1]. If \(V = \mathbb{S}^{2}\), then the connected component of the identity coincides with the arcwise connected component of the identity (see [6, p. 1]).
- 20.
Actually, even {0} could not be left fixed by \(\tilde{h}_{t}\).
- 21.
Making use of (2.4), we define R(⋅ , t) as follows. Let us identify \(\mathbb{S}^{2}\) with the unit sphere in \(\mathbb{R}^{3}\) and endow it with parallels and meridians with the north pole identified with ∞ and \(0 \in \mathbb{S}^{2}\) identified with the south pole. The stereographic projection from the north pole to the tangent plane at the south pole provides an identification of points of \(\mathbb{R}^{2}\) with points of \(\mathbb{S}^{2}\setminus \{\infty \}\) (we employ a scale reduction of a factor \(2\) on the stereographic projection so that the equator is mapped onto the unit circle of \(\mathbb{R}^{2}\)). Suppose first that \(\tilde{h}_{t}(0)\) is not the south pole. Let P(t) be the intersection between the equator of \(\mathbb{S}^{2}\) and the meridian passing through (the poles and) \(\tilde{h}_{t}(0)\). Let r(t) be the line (in \(\mathbb{R}^{3}\)) joining p(t) to q(t), where p(t) [respectively q(t)] is the point on the equator having the longitude of P(t) plus (respectively minus) π∕2. Then R(⋅ , t) is the (smallest) rotation around r(t) sending \(\tilde{h}_{t}(0)\) into the origin. This is a rotation of angle given by the latitude of \(\tilde{h}_{t}(0)\) plus π∕2, clearly this rotation takes the above-mentioned meridian into itself. If \(\tilde{h}_{t}(0)\) is the south pole, we define \(R(\cdot,t):=\mathrm{ id}_{\mathbb{S}^{2}}(\cdot )\). Then, \(R(\cdot,t) \in \mathrm{ Diff}^{+}(\mathbb{S}^{2})\), and R is of class \(\mathcal{C}^{\infty }\) and satisfies the required properties.
- 22.
- 23.
- 24.
See, e.g., [4].
- 25.
Notice that \(\mathfrak{m}\) can be thought of as an element of \(\mathrm{Diff}_{\mathrm{c}}(\mathbb{R}^{2})\).
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Bellettini, G., Beorchia, V., Paolini, M., Pasquarelli, F. (2015). Stable Maps and Morse Descriptions of an Apparent Contour. In: Shape Reconstruction from Apparent Contours. Computational Imaging and Vision, vol 44. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-662-45191-5_2
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