FOCAL SURFACES OF FRONTS ASSOCIATED TO UNBOUNDED PRINCIPAL CURVATURES KEISUKE TERAMOTO

2025-04-27 0 0 603.23KB 21 页 10玖币
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FOCAL SURFACES OF FRONTS ASSOCIATED TO UNBOUNDED
PRINCIPAL CURVATURES
KEISUKE TERAMOTO
Abstract. We study focal surfaces of (wave) fronts associated to unbounded principal
curvatures near non-degenerate singular points of initial fronts. We give characterizations
of singularities of those focal surfaces in terms of types of singularities and geometrical
properties of initial fronts. Moreover, we investigate behavior of the Gaussian curvature
of the focal surface.
1. Introduction
Focal surfaces (or caustics) of regular surfaces in the Euclidean 3-space R3can be
characterized by several ways: loci of centers of principal curvature spheres of initial
surfaces, singular value sets of the normal congruence, and the bifurcation set of the family
of distance squared functions for instance (cf. [4,12,15,28]). Although initial surfaces have
no singular points, their focal surfaces have singularities in general ([1,2,12,28,29,37]). It
is known that types of (co)rank one singularities on focal surfaces correspond to geometric
properties arising from principal curvatures of initial surfaces ([12,28,29]). Investigating
singularities of focal surfaces, one might have new geometrical properties of surfaces
([3–5,12,23]).
On the other hand, there are classes of surfaces with singular points called frontals and
(wave)fronts. These surfaces admit smooth unit normal vector field even at singular points.
Owing to this property, frontals and fronts can be considered as a kind of generalization
of regular surfaces (i.e., immersions) in R3. Recently, there are several studies on frontals
and fronts from differential geometric viewpoint, and various geometric invariants at
singular points are introduced ([6–11,16,20,21,25–27,34]). Moreover, relation among
behavior of the curvatures of frontals and fronts and geometric invariants are considered
([11,21,32,34,36]). In particular, for a front, it is known that one principal curvature can
be extended as a bounded 𝐶∞function near a non-degenerate singular point, and another
is unbounded near such a singular point ([22,24,36]). Although one principal curvature
of a front is unbounded near a non-degenerate singular point, the radius function (i.e., the
reciprocal) of it can be extended as a 𝐶∞function near the singular point. This fact plays
a crucial role to study focal surfaces of fronts with bounded Gaussian curvature because
the bounded principal curvature vanishes on the set of singular points of such fronts, and
hence we cannot define corresponding focal surfaces near such points (cf. [17,19,24,30]).
In this paper, we investigate singularities and geometrical properties of focal surfaces
of fronts associated to unbounded principal curvatures. We note that characterizations
of singularities on focal surfaces relative to bounded principal curvatures are given in
[35]. Moreover, if the initial front has a cuspidal edge, then the focal surface associated
Date: October 13, 2022.
2020 Mathematics Subject Classification. 53A05, 53A55, 57R45.
Key words and phrases. Front, Focal surface, Singularity, Principal curvature, Sub-parabolic point,
Gaussian curvature.
The author was partially supported by JSPS KAKENHI Grant Numbers JP19K14533, JP22K13914 and
JP20H01801, and the Japan-Brazil bilateral project JPJSBP1 20190103.
1
arXiv:2210.06221v1 [math.DG] 12 Oct 2022
2 K. TERAMOTO
to the unbounded principal curvature is regular at the corresponding point ([35]). Thus
we consider focal surfaces of fronts with singular points of the second kind, which are
classes of non-degenerate singular points of fronts. A swallowtail singularity is a typical
example. To give characterizations of singularities of the focal surface, we recall behavior
of principal curvatures and principal vectors around a non-degenerate singular point in
Section 3. By observing the unbounded principal curvature and corresponding principal
vector, we extend a concept of sub-parabolic points to singular points of the second kind
on fronts in Section 4. In particular, we show relation between behavior of the Gaussian
curvature and sub-parabolic points of a front (Propositions 4.3 and 4.4).
In Section 5, we study a focal surface of a front associated to the unbounded principal
curvature. We give characterizations of singularities on focal surfaces by geometrical
properties of initial fronts (Theorem 5.6 and Proposition 5.9). Furthermore, we study
contact between singular sets of the initial front and the focal surface associated to the
unbounded principal curvature. We characterize types of singularities of the focal surface
in terms of the contact order of these curves (Proposition 5.14). Finally, we observe that
the behavior of the Gaussian curvature of the focal surface. Especially, for the case of a
singular point of the second kind, we give characterization for the rational boundedness of
the Gaussian curvature of the focal surface by a certain geometrical property of the initial
front (Theorem 5.15).
2. Preliminaries
We recall some notions on fronts. For details, see [1,2,6,12,18,34].
2.1. Fronts. Let (Σ;𝑢, 𝑣)be a domain in the (𝑢, 𝑣)-plane R2. Let 𝑓:Σ→R3be a
𝐶∞map. Then 𝑓is said to be a frontal if there exists a 𝐶∞map 𝜈:Σ→S2such that
𝑑𝑓𝑞(𝑿), 𝜈(𝑞)=0for any 𝑞∈Σand 𝑿∈𝑇𝑞Σ, where S2is the standard unit sphere in
R3and h·,·iis the canonical inner product on R3. Moreover, a frontal 𝑓is a front if the
pair (𝑓 , 𝜈):Σ→R3×S2gives an immersion. We call 𝜈aunit normal vector field or a
Gauss map of 𝑓.
We fix a frontal 𝑓. A point 𝑝∈Σis called a singular point of 𝑓if 𝑓is not an immersion
at 𝑝. We denote by 𝑆(𝑓)the set of singular points of 𝑓. Set a function 𝜆:Σ→Rby
(2.1) 𝜆(𝑢, 𝑣)=det(𝑓𝑢, 𝑓𝑣, 𝜈)(𝑢, 𝑣) ( 𝑓𝑢=𝜕 𝑓 /𝜕𝑢, 𝑓𝑣=𝜕 𝑓 /𝜕𝑣),
where det is the determinant of 3×3matrices. Then one can check that 𝜆−1(0)=𝑆(𝑓)
holds by the definition. We call the function 𝜆the signed area density function of 𝑓.
Take a singular point 𝑝∈𝑆(𝑓)of a frontal 𝑓. Then 𝑝is said to be non-degenerate
(resp. degenerate) if (𝜆𝑢(𝑝), 𝜆𝑣(𝑝)) ≠(0,0)(resp. (𝜆𝑢(𝑝), 𝜆𝑣(𝑝)) =(0,0)) holds. For a
non-degenerate singular point 𝑝of 𝑓, there exist an open neighborhood 𝑈(⊂ Σ)of 𝑝and
a regular curve 𝛾:(−𝜀, 𝜀) 3 𝑡↦→ 𝛾(𝑡) ∈ 𝑈(𝜀 > 0)such that 𝛾(0)=𝑝and 𝜆(𝛾(𝑡)) =0
on 𝑈by the implicit function theorem. Moreover, since a non-degenerate singular point
𝑝is a corank one singular point (i.e., rank 𝑑𝑓𝑝=1), there exists a never-vanishing vector
field 𝜂on 𝑈such that 𝑑𝑓𝑞(𝜂𝑞)=0for any 𝑞∈𝑆(𝑓) ∩ 𝑈(𝜂𝑞∈𝑇𝑞𝑈). We call 𝛾and 𝜂the
singular curve for 𝑓and a null vector field, respectively. We remark that one can take a
null vector field 𝜂of a front near a corank one singular point 𝑝.
A non-degenerate singular point 𝑝is said to be of the first kind if 𝛾0=𝑑𝛾/𝑑𝑡 and 𝜂are
linearly independent at 𝑝=𝛾(0). Otherwise, it is said to be of the second kind. Let 𝑝be
a singular point of the second kind of a frontal 𝑓. Then 𝑝is said to be admissible if for
each open neighborhood 𝑈of 𝑝, the intersection 𝑆(𝑓) ∩ 𝑈contains a singular point of the
first kind. Otherwise, we call 𝑝non-admissible. We say that a non-degenerate singular
point 𝑝is admissible if 𝑝is either of the first kind or the admissible second kind.
FOCAL SURFACES OF FRONTS 3
Definition 2.1. Let 𝑓:(Σ, 𝑝) → R3be a 𝐶∞map germ and 𝑝a singular point of 𝑓. Then
(1) 𝑓is a cuspidal edge at 𝑝if 𝑓is A-equivalent to the germ (𝑢, 𝑣) ↦→ (𝑢, 𝑣2, 𝑣3)at
the origin.
(2) 𝑓is a swallowtail at 𝑝if 𝑓is A-equivalent to the germ (𝑢, 𝑣) ↦→ (𝑢, 4𝑣3+2𝑢𝑣, 3𝑣4+
𝑢𝑣2)at the origin.
(3) 𝑓is a cuspidal butterfly at 𝑝if 𝑓is A-equivalent to the germ (𝑢, 𝑣) ↦→ (𝑢, 5𝑣4+
2𝑢𝑣, 4𝑣5+𝑢𝑣2)at the origin.
(4) 𝑓is a cuspidal lips at 𝑝if 𝑓is A-equivalent to the germ (𝑢, 𝑣) ↦→ (𝑢, 3𝑣4+
2𝑢2𝑣2, 𝑣3+𝑢2𝑣)at the origin.
(5) 𝑓is a cuspidal beaks at 𝑝if 𝑓is A-equivalent to the germ (𝑢, 𝑣) ↦→ (𝑢, 3𝑣4−
2𝑢2𝑣2, 𝑣3−𝑢2𝑣)at the origin.
Here, two map germs 𝑓1, 𝑓2:(R2,0)→(R3,0)are A-equivalent if there exist diffeomor-
phism germs 𝜑:(R2,0) → (R2,0)on the source and Φ:(R3,0) → (R3,0)on the target
such that 𝑓2= Φ ◦𝑓1◦𝜑−1.
A cuspidal edge, a swallowtail and a cuspidal butterfly are non-degenerate front singular
points. Moreover, a cuspidal edge and a swallowtail are generic singularities of fronts in
R3, and a cuspidal lips/beaks and a cuspidal butterfly are generic corank one singularities
of one-parameter bifurcation of fronts (cf. [2,37]). Further, a cuspidal edge is a singular
point of the first kind, and a swallowtail and a cuspidal butterfly are of the admissible
second kind (cf. [21]). For these singular points, the following characterizations are
known.
Fact 2.2 ([13,14,18,33]).Let 𝑓:Σ→R3be a front and 𝑝∈Σa corank one singular
point of 𝑓. Then we have the following.
(1) 𝑓is a cuspidal edge at 𝑝if and only if 𝜂𝜆(𝑝)≠0.
(2) 𝑓is a swallowtail at 𝑝if and only if 𝑑𝜆(𝑝)≠0,𝜂𝜆(𝑝)=0and 𝜂𝜂𝜆(𝑝)≠0.
(3) 𝑓is a cuspidal butterfly at 𝑝if and only if 𝑑𝜆(𝑝)≠0,𝜂𝜆(𝑝)=𝜂𝜂𝜆(𝑝)=0and
𝜂𝜂𝜂𝜆(𝑝)≠0.
(4) 𝑓is a cuspidal lips at 𝑝if and only if 𝑑𝜆(𝑝)=0and det(H𝜆)( 𝑝)>0, that is, 𝜆
has a Morse type singularity of index zero or two at 𝑝.
(5) 𝑓is a cuspidal beaks at 𝑝if and only if 𝑑𝜆(𝑝)=0,𝜂𝜂𝜆(𝑝)≠0and det(H𝜆)( 𝑝)<
0, that is, 𝜆has a Morse type singularity of index one and 𝜂𝜂𝜆 ≠0at 𝑝.
Here, 𝜆is the signed area density function of 𝑓as in (2.1),𝜂is a null vector field, 𝜂𝜆
means the directional derivative of 𝜆in the direction 𝜂and det(H𝜆)is the Hessian of 𝜆.
2.2. Geometric invariants. We recall geometric invariants of fronts.
2.2.1. Geometric invariants of cuspidal edges. First we consider the case of cuspidal
edges. Let 𝑓:Σ→R3be a front, 𝜈its Gauss map and 𝑝a cuspidal edge of 𝑓. Let 𝛾(𝑡)
be a singular curve passing through 𝑝and 𝜂a null vector field of 𝑓. Then one can define
the following geometric invariants: the singular curvature 𝜅𝑠([34]), the limiting normal
curvature 𝜅𝜈([21,34]), the cuspidal curvature 𝜅𝑐([21]) and the cuspidal torsion 𝜅𝑡([20]).
We note that 𝜅𝑠is an intrinsic invariant and its sing has a geometrical meaning (see [9,34]).
It is known that 𝜅𝑐does not vanish when 𝛾consists of cuspidal edges ([20,21]). We remark
that these invariants can be defined at singular points of the first kind for frontals but not
fronts. In such cases, 𝜅𝑐vanishes at non-front singular points (cf. [21, Proposition 3.11]).
The limiting normal curvature 𝜅𝜈relates to the boundedness of the Gaussian curvature
of a front with a cuspidal edge. In fact, the Gaussian curvature 𝐾of a front 𝑓is bounded
on a sufficiently small neighborhood 𝑈of a cuspidal edge 𝑝if and only if 𝜅𝜈vanishes
along the singular curve 𝛾through 𝑝([21, Theorem 3.9]). In this case, 𝐾can be extended
as a 𝐶∞function on 𝑈. Moreover, the following property holds.
4 K. TERAMOTO
Fact 2.3 ([21, Remark 3.19],[7, Theorem 1.9]).Let 𝑓be a front in R3and 𝑝a cuspidal
edge. Let 𝐾be the Gaussian curvature of 𝑓defined on the set of regular points of 𝑓.
Suppose that 𝐾is bounded on a sufficiently small neighborhood 𝑈of 𝑝. Then 𝐾satisfies
(2.2) 4𝐾=−4𝜅2
𝑡−𝜅𝑠𝜅2
𝑐
at 𝑝.
On the other hand, we can take a coordinate system (𝑈;𝑢, 𝑣)around 𝑝satisfying the
following properties:
(1) the 𝑢-axis is the singular curve,
(2) 𝜕𝑣gives a null vector field, and
(3) there are no singular points other than the 𝑢-axis.
We call this local coordinate system (𝑈;𝑢, 𝑣)adapted ([18,21,34]). Moreover, an adapted
coordinate system (𝑈;𝑢, 𝑣)is called special adapted if the frame {𝑓𝑢, 𝑓𝑣𝑣, 𝜈}gives an
orthonormal frame along the 𝑢-axis in addition to the above conditions ([34, Lemma 3.2],
[21]).
On an adapted coordinate system (𝑈;𝑢, 𝑣), since 𝜂=𝜕𝑣,𝑓𝑣(𝑢, 0)=0holds. Thus there
exists a 𝐶∞map 𝑔:𝑈→R3such that 𝑓𝑣=𝑣𝑔. We note that 𝑔does not vanish along the
𝑢-axis since 𝑓𝑣𝑣 =𝑔holds along the 𝑢-axis. Therefore the pair {𝑓𝑢, 𝑔, 𝜈}gives a frame
along 𝑓. Moreover, when we take a special adapted coordinate system (𝑈;𝑢, 𝑣)around a
cuspidal edge 𝑝, then {𝑓𝑢, 𝑔, 𝜈}gives an orthonormal frame along the 𝑢-axis. In particular,
𝜈can be taken as 𝜈=(𝑓𝑢×𝑔)/| 𝑓𝑢×𝑔|. Using these mappings, we define the following
functions:
e
𝐸=h𝑓𝑢, 𝑓𝑢i,e
𝐹=h𝑓𝑢, 𝑔i,e
𝐺=h𝑔, 𝑔i,
e
𝐿=−h𝑓𝑢, 𝜈𝑢i,e
𝑀=−h𝑔, 𝜈𝑢i,e
𝑁=−h𝑔, 𝜈𝑣i.
(2.3)
We remark that e
𝐸e
𝐺−e
𝐹2>0on 𝑈. Relation between geometric invariants stated above
and the functions in (2.3) are known ([9,36]).
2.2.2. Geometric invariants at singular points of the second kind. We next consider
geometric invariants at a singular point of the admissible second kind of a front. Let 𝑝be
a singular point of the second kind of a front 𝑓in R3. Then one can take a local coordinate
system (𝑈;𝑢, 𝑣)centered at 𝑝satisfying
(1) 𝑓𝑢(𝑝)=0,
(2) the 𝑢-axis is the singular curve on 𝑈, and
(3) |𝑓𝑣(𝑝)| =1.
We also call this local coordinate system adapted. Further, if an adapted coordinate
system (𝑈;𝑢, 𝑣)around a singular point of the second kind 𝑝satisfies h𝑓𝑢𝑣, 𝑓𝑣i(𝑝)=0,
then (𝑈;𝑢, 𝑣)is called a strongly adapted coordinate system (cf. [21, Definitions 4.1 and
4.6]).
Using an adapted coordinate system (𝑈;𝑢, 𝑣)around a singular point of the second kind
𝑝, we define geometric invariants at 𝑝as follows:
(2.4) 𝜅𝜈(𝑝)=lim
𝑢→0
h𝑓𝑢𝑢, 𝜈i
|𝑓𝑢|2(𝑢, 0), 𝜇𝑐=−h𝑓𝑢𝑣, 𝜈𝑢i
|𝑓𝑢𝑣 ×𝑓𝑣|2(𝑝).
𝜅𝜈(𝑝)is the limiting normal curvature and 𝜇𝑐is the normalized cuspidal curvature at
𝑝(see [21, Proposition 2.9 and (4.7)]). These invariants are related to the boundedness
of the Gaussian and the mean curvature near singular points of the second kind (see
[21, Propositions 4.2 and 4.3, Theorem 4.4])
FOCAL SURFACES OF FRONTS 5
Let us take an adapted coordinate system (𝑈;𝑢, 𝑣)around a singular point of the
(admissible) second kind 𝑝. Then one can take a null vector field 𝜂as
𝜂=𝜕𝑢+𝑒(𝑢)𝜕𝑣,
where 𝑒(𝑢)is a 𝐶∞function with 𝑒(0)=0([21]). We note that there exists a positive
integer 𝑙such that 𝑒(0)=𝑒0(0)=· · · =𝑒(𝑙−1)(0)=0and 𝑒(𝑙)(0)≠0if 𝑝is of the
admissible (cf. [31, Lemma 2.2]).
Since 𝑑𝑓 (𝜂)=𝜂 𝑓 =𝑓𝑢+𝑒(𝑢)𝑓𝑣=0along the 𝑢-axis, there exists a 𝐶∞map ℎ:𝑈→R3
such that 𝜂 𝑓 =𝑣ℎ, and hence 𝑓𝑢=𝑣ℎ −𝑒(𝑢)𝑓𝑣. Using this map ℎ, we have 𝜆=
det(𝑓𝑢, 𝑓𝑣, 𝜈)=𝑣det(ℎ, 𝑓𝑣, 𝜈). By the non-degeneracy, 𝜆𝑣(𝑝)≠0holds. Thus we have
det(ℎ, 𝑓𝑣, 𝜈) (𝑝)≠0. This implies that {ℎ, 𝑓𝑣, 𝜈}gives a frame along 𝑓near 𝑝. We take 𝜈
satisfying 𝜆𝑣(𝑝)>0, so one may take 𝜈as
𝜈=ℎ×𝑓𝑣
|ℎ×𝑓𝑣|
in what follows.
We define the following functions:
b
𝐸=hℎ, ℎi,b
𝐹=hℎ, 𝑓𝑣i,b
𝐺=h𝑓𝑣, 𝑓𝑣i,
b
𝐿=−hℎ, 𝜈𝑢i,b
𝑀=−hℎ, 𝜈𝑣i,b
𝑁=−h𝑓𝑣, 𝜈𝑣i.
(2.5)
We note that b
𝐸b
𝐺−b
𝐹2>0on 𝑈. Using functions as in (2.5), differentials 𝜈𝑢and 𝜈𝑣of 𝜈
can be written as follows.
Lemma 2.4 ([36, Lemma 2.8]).On an adapted coordinate system (𝑈;𝑢, 𝑣),𝜈𝑢and 𝜈𝑣can
be written as
𝜈𝑢=b
𝐹(𝑣b
𝑀−𝑒(𝑢)b
𝑁) − b
𝐺b
𝐿
b
𝐸b
𝐺−b
𝐹2ℎ+b
𝐹b
𝐿−b
𝐸(𝑣b
𝑀−𝑒(𝑢)b
𝑁)
b
𝐸b
𝐺−b
𝐹2𝑓𝑣,
𝜈𝑣=b
𝐹b
𝑁−b
𝐺b
𝑀
b
𝐸b
𝐺−b
𝐹2ℎ+b
𝐹b
𝑀−b
𝐸b
𝑁
b
𝐸b
𝐺−b
𝐹2𝑓𝑣.
(2.6)
If 𝑓is a front at a singular point of the second kind 𝑝, then 𝜂𝜈 ≠0along the singular
curve 𝛾. We rephrase this condition using functions as in (2.5). By 𝜂=𝜕𝑢+𝑒(𝑢)𝜕𝑣and
Lemma 2.4, it follows that
(2.7) 𝑑𝜈(𝜂)=𝜈𝑢+𝑒(𝑢)𝜈𝑣=−b
𝐿+𝑒(𝑢)b
𝑀
b
𝐸b
𝐺−b
𝐹2(b
𝐺ℎ −b
𝐹 𝑓𝑣)
holds along the 𝑢-axis. Thus if 𝑓is a front around 𝑝, then b
𝐿+𝑒(𝑢)b
𝑀≠0along the 𝑢-axis,
in particular, b
𝐿(𝑝)≠0. Moreover, we have the following.
Lemma 2.5 ([21,36]).Take a strongly adapted coordinate system (𝑈;𝑢, 𝑣)around a
singular point of the (admissible)second kind 𝑝of a front 𝑓in R3. Then 𝜅𝜈(𝑝)=b
𝑁(𝑝)
and 𝜇𝑐=b
𝐿(𝑝)/ b
𝐸(𝑝)hold.
Proof. For 𝜅𝜈, it follows from [36, Lemma 2.9]. For 𝜇𝑐, we have the expression by (2.5)
and [21, (4.13)].
2.2.3. Rational boundedness of the Gaussian curvature. Let 𝑓:Σ→R3be a front
and 𝑝∈Σa non-degenerate singular point of 𝑓. Then the Gaussian curvature 𝐾of
𝑓is unbounded near 𝑝in general. In [21], a notion of the rational boundedness for
unbounded functions was introduced. (For precise definition and descriptions of the
rational boundedness, see [21, Definition 3.4 and Page 260].) For the Gaussian curvature
of a front, the following assertion holds.
摘要：

FOCALSURFACESOFFRONTSASSOCIATEDTOUNBOUNDEDPRINCIPALCURVATURESKEISUKETERAMOTOAbstract.Westudyfocalsurfacesof(wave)frontsassociatedtounboundedprincipalcurvaturesnearnon-degeneratesingularpointsofinitialfronts.Wegivecharacterizationsofsingularitiesofthosefocalsurfacesintermsoftypesofsingularitiesandgeo...

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分类：图书资源 价格：10玖币 属性：21 页 大小：603.23KB 格式：PDF 时间：2025-04-27

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