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Complete flat fronts as hypersurfaces in Euclidean space.

1. Introduction. Let [R.sup.n+1] be the Euclidean (n + 1)-space. By Hartman Nirenberg's theorem [2], any complete flat hypersurface in [R.sup.n+1] must be a cylinder over a plane curve. Here, a cylinder is a regular hypersurface which is congruent to f : [R.sup.n] [right arrow] [R.sup.n+1] defined by

f (t, [w.sub.2], ..., [w.sub.n]) := (x(t),y(t),[w.sub.2], ..., [w.sub.n]),

where t [right arrow] (x(t),y(t)) is a regular curve in [R.sup.2]. We remark that Massey [8] gave an alternative proof for n = 2.

However, in [R.sup.3], there are non-trivial flat surfaces with admissible singularities called flat fronts. Here, a front is a generalized notion of regular surfaces (more generally, regular hypersurfaces) with admissible singular points. See Section 2 for precise definitions. Murata Umehara [9] gave a representation formula for complete flat fronts with non-empty singular set, and proved the four vertex type theorem: Let [xi] : [S.sup.1] [right arrow] [S.sup.2] be a regular curve without inflection points, and [alpha] = a(t)dt a 1-form on [S.sup.1] = R/2[pi]Z such that [mathematical expression not reproducible] holds. Then, f[epsilon],[alpha]a: [S.sup.1] x R [right arrow] [R.sup.3] defined by

(1.1) [mathematical expression not reproducible]

is a complete flat front with non-empty singular set.

Conversely, let f : [M.sup.2] [right arrow] [R.sup.3] be a complete flat front defined on a connected smooth 2-manifold [M.sup.2]. If the singular set S(f) off is not empty, then f is umbilic-free, co-orientable, [M.sup.2] is diffeomorphic to [S.sup.1] x R, and f is given by (1.1). Moreover, if the ends of f are embedded, f has at least four singular points other than cuspidal edges.

Therefore, it is natural to ask what occurs in the higher dimensional cases. In this paper, we prove that there do not exist any non-trivial flat fronts in higher dimensions:

Theorem 1. If n [greater than or equal to] 3, there do not exist any complete flat fronts with non-empty singular set in [R.sup.n+1].

Combining Hartman Nirenberg's theorem [2], Murata Umehara's theorem [9] and Theorem 1, we have the classification of complete flat fronts in [R.sup.n+1].

We remark that, although there do not exist any complete flat fronts in [R.sup.n+1] (n [greater than or equal to] 3), there are many weakly complete ones. For example, we can construct a weakly complete flat front by a pair ([gamma](t),a(t)) of a complete regular curve [gamma](t) in [S.sup.n] and a smooth function a(t) on R (cf. Proposition 8). Here, we denote by Sn the n-sphere of constant sectional curvature 1. Moreover, by a regular curve in [R.sup.n+1], one may construct a flat front called tangent developable. (See [7] for more details and properties of singularities of tangent developables.)

We also remark that there are several works related to Murata Umehara's theorem. Naokawa [10] gave an estimation of singular points other than cuspidal edges on asymptotic completions of developable Mobius strips. On the other hand, flat fronts can be considered as fronts with one principal curvature zero. In a previous paper [5], the author gave a classification of weakly complete fronts with one principal curvature non-zero constant.

With respect to the case of non-flat ambient spaces, it is known that flat fronts in [R.sup.n+1] are identified with fronts of constant sectional curvature 1 (CSC-1 fronts) in [S.sup.n+1] via the central projection of a hemisphere to a tangent space. Therefore, the local nature of flat fronts in [R.sup.n+1] is the same as that of CSC-1 fronts in [S.sup.n+1]. However, they may display different global properties. In [6], the author gave a classification of complete CSC-1 fronts, which is a generalization of O'Neill Stiel's theorem [11]. In particular, in the case of n [greater than or equal to] 3, there exist many non-trivial complete CSC-1 fronts in [S.sup.n+1], although there do not exist any complete flat fronts other than cylinders in [R.sup.n+1]. (See also [4] for the case of negative sectional curvature.)

This paper is organized as follows: In Section 2, we shall review the definition and fundamental properties of flat fronts. Using them, we shall prove Theorem 1 in Section 3.

2. Preliminaries. We denote by [R.sup.n+1] the Euclidean (n + 1)-space, and Sn the unit sphere

[S.sup.m] :={x [member of] [R.sup.n+1]; x x x = 1},

where the dot '.'is the canonical inner product on

[R.sup.n+1]. Let [M.sup.n] be a connected smooth n-manifold and

f : [M.sup.n] [right arrow] [R.sup.n+1]

a smooth map. A point p [member of] [M.sup.n] is called a singular point if f is not an immersion at p. Otherwise, we call p a regular point. Denote by S(f) ([subset] [M.sup.n]) the set of singular points. If S(f) is empty, we call f a (regular) hypersurface.

A smooth map f : [M.sup.n] [right arrow] [R.sup.n+1] is called a frontal, if for each point p [member of] [M.sup.n], there exist a neighborhood U of p and a smooth map v : U [right arrow] [S.sup.n] such that

d[f.sub.q](v) x v(q) = 0

holds for each q [member of] U and v [member of] [T.sub.q][M.sup.n]. Such a v is called the unit normal vector field or the Gauss map of f. If v can be defined throughout [M.sup.n], f is called co-orientable. On the other hand, we say orientable if [M.sup.n] is orientable. If

(L :=) (f, v) : U [right arrow] [R.sup.n+1] x [S.sup.n]

gives an immersion, f is called a wave front (or a front, for short). The map L = (f, v) is called the Legendrian lift of f.

2.1. Completeness, weak completeness, umbilic points. The first fundamental form (i.e., the induced metric) is given by [ds.sup.2] := df x df. For a front f : [M.sup.n] [right arrow] [R.sup.n+1] with a (possibly locally defined) unit normal vector field v,

[ds.sup.2.sub.#] := [ds.sup.2] + dv x dv

gives a positive definite Riemannian metric called the lift metric. If the lift metric [ds.sup.2.sub.#] is complete, f is called weakly complete. On the other hand, f is called complete, if there exists a symmetric covariant (0, 2)-tensor T on [M.sup.n] with compact support such that [ds.sup.2] + T gives a complete metric on [M.sup.n]. In this case, the singular set S(f) must be compact. As noted in [9], if S(f) is empty, then f : [M.sup.n] [right arrow] [R.sup.n+1] is complete as a front if and only if f is complete as a regular hypersurface (i.e., ([M.sup.n], [ds.sup.2]) is a complete Riemannian manifold).

Fact 2 ([9, Lemma 4.1]). A complete front is weakly complete.

A point p [member of] [M.sup.n] is called an umbilic point, if there exist real numbers [[delta].sup.1], [[delta].sup.2] such that

[[delta].sup.1] [(df).sup.p] = [[delta].sup.2] [(dv).sub.p], ([[delta].sup.1], [[delta].sup.2]) [not equal to] (0, 0)

hold. For a positive number [delta] > 0, set

(2.1) [f.sup.[delta]] := f + [delta]v, [v.sup.[deta]] := v.

Then we can check that [f.sup.[delta]] is a front and [v.sup.[delta]] gives a unit normal along [f.sup.[delta]]. Such an [f.sup.[delta]] is called the parallel front of f. Umbilic points are common in its parallel family.

Fact 3 ([6, Lemma 2.7]). Let p [member of] [M.sup.n] be a singular point of a front f. Then, p is an umbilic point if and only if rank[(df).sub.p] = 0 holds. In this case, we have rank[(dv).sub.p] = n for any unit normal vector field of f.

2.2. Flat fronts. In [12,14], Saji Umehara Yamada introduced coherent tangent bundles, which is a generalized notion of Riemannian manifolds.

Let [epsilon] be a vector bundle of rank n over a smooth n-manifold [M.sup.n]. We equip a fiber metric (,) on [epsilon] and a metric connection D on ([epsilon], (,)). Let [phi] : [TM.sup.n] [right arrow] [epsilon] be a bundle homomorphism such that

(2.2) [D.sub.X][phi](Y)- [D.sub.Y][phi](X)- [phi]([X, Y]) = 0

holds for arbitrary smooth vector fields X, Y on [M.sup.n]. Then

[epsilon] = ([epsilon], (,), D, [alpha])

is called a coherent tangent bundle over [M.sup.n].

It is known that, in general, coherent tangent bundles can be constructed only from positive semi-definite metrics called Kossowski metrics (cf. [3,15]).

We shall review the coherent tangent bundles induced from frontals (cf. [14, Example 2.4]). For a frontal f : [M.sup.n] [right arrow] [R.sup.n+1], set [E.sup.f], [(,).sub.f], [D.sub.f] and [[phi].sub.f], respectively, as follows:

* [E.sub.f] is the subbundle of the pull-back bundle f *T[R.sup.n+1] perpendicular to v,

* [(,).sub.f] is the metric on [[epsilon].sub.f] induced from the canonical metric on [R.sup.n+1],

* [D.sup.f] is the tangential part of the Levi Civita connection on [R.sup.n+1],

* [[phi].sub.f]T[M.sub.n] [right arrow] [[epsilon].sub.f] defined as (X):= df (X).

Then, [[epsilon].sub.f] = ([epsilon].sub.f], [(,).sub.f], [D.sub.f], [[phi].sub.f]) is a coherent tangent bundle, which we call the induced coherent tangent bundle.

Definition 4 ([6]). A coherent tangent bundle is said to be flat if

[R.sup.D] (X, Y) [xi] = 0

holds for all smooth vector fields X, Y on [M.sup.n] and each smooth section [xi] of [epsilon], where [R.sup.D] is the curvature tensor of the connection D given by

[R.sup.D](X, Y)[xi] := [D.sub.X][D.sub.Y][xi] - [D.sub.Y][D.sub.X][xi] - [D.sub.[X,Y]][xi].

A frontal f is called flat, if the induced coherent tangent bundle [[epsilon].sub.f] is flat.

In [6], the following characterization of flatness was proved by using the Gauss equation for frontals given by Saji Umehara Yamada [13, Proposition 2.4].

Fact 5 ([6, Lemma 3.3]). Let f : Mn [right arrow] [R.sup.n+1] be a frontal with a unit normal vector field v. Then f is flat if and only if

(2.3) rank(dv) [less than or equal to] 1

holds on [M.sup.n].

We remark that Murata Umehara [9] defined the flatness for frontals in [R.sup.3] by the condition (2.3). Therefore, our definition of flatness is compatible to that given by Murata Umehara. We also remark that, if S(f) is empty, then f : [M.sup.n] [right arrow] [R.sup.n+1] is flat as a front if and only if f is flat as a regular hypersurface.

3. Proof of Theorem 1. Denote by [U.sub.f] the set of umbilic points. Since [U.sub.f] is a closed subset in [M.sup.n], the non-umbilic point set [M.sup.n] \ [U.sub.f] is open.

Lemma 6. Let f : [M.sup.n] [right arrow] [R.sup.n+1] be a nontotally-umbilic flat front. For each non-umbilic point q [member of] [M.sup.n] \ [U.sub.f], there exist a local coordinate neighborhood (U; [u.sub.1],..., [u.sub.n]) of q and a smooth function p = p ([u.sub.1],..., [u.sub.n]) on U such that

(3.1) [mathematical expression not reproducible]

hold for each j = 2,..., n, and {[mathematical expression not reproducible]} is a frame on U. For each u1, set the slice [mathematical expression not reproducible] of U as

[mathematical expression not reproducible].

Then, the restriction [mathematical expression not reproducible] is a totally geodesic embedding for each [u.sub.1].

Proof. Since f is flat, Fact 5 implies that there exists a local coordinate system (V; [v.sub.1], ..., [v.sub.n]) around q [member of] [M.sup.n] \ [U.sub.f] such that

[mathematical expression not reproducible]

holds. Then, we have

[mathematical expression not reproducible].

Since f is a front, {[mathematical expression not reproducible]} is linearly independent around q. And hence, we have that {[mathematical expression not reproducible]} or {[mathematical expression not reproducible]} is linearly independent around q. In each case, there exists a positive number [delta] > 0 such that the parallel front [f.sup.][delta] := f + [delta]v is a flat immersion around q (cf. (2.1)). Since f is umbilic-free around q, so is [f.sup.[delta]]. Let (U; [u.sub.1],...,[u.sub.n]) be a curvature line coordinate system of [f.sup.[delta]] around q [member of] [M.sup.n]\ [U.sub.f]. That is, for each j = 2...

(3.2) [mathematical expression not reproducible]

and

(3.3) [mathematical expression not reproducible]

hold, where [alpha] = [alpha] ([u.sub.1], ..., [u.sub.n]) is a smooth function on U. In this case, the principal curvatures [[lambda].sup.[delta].sub.1], ..., [[lambda].sup.[delta].sub.n] of [[lambda].sup.[delta]] are given by [alpha] [not equal to] a, AS = 0 (j = 2, ...,n). Since fS is umbilic-free, a = 0 on U. Substituting (2.1) into Eqs. (3.2) and (3.3), we may conclude that (3.1) holds with

[rho]:= 1 + [delta][alpha]/[alpha].

With respect to the third assertion, [mathematical expression not reproducible] gives a unit normal vector field of [mathematical expression not reproducible]. Then, for each j = 2, ..., n,

[mathematical expression not reproducible]

and[mathematical expression not reproducible]. Together with [mathematical expression not reproducible] on [mathematical expression not reproducible], we have the conclusion.

By Lemma 6, since the image of [mathematical expression not reproducible] is included in an (n--1)-dimensional affine subspace [mathematical expression not reproducible] of [R.sup.n+1] for each [u.sub.1], by a coordinate change of ([u.sub.2], ..., [u.sub.n]), we may take a new coordinate system (U'; [u.sub.1], [w.sub.2], ..., [w.sub.n]) such that ([w.sub.2], ..., [w.sub.n]) is the canonical Euclidean coordinate system of [mathematical expression not reproducible] for each [u.sub.1]. Namely, [mathematical expression not reproducible] holds for j, k = 2, ..., n.

[mathematical expression not reproducible]

respectively. Then, we have

f ([u.sub.1], [w.sub.2], ..., [w.sub.n])

= [sigm]([u.sub.1])+ [w.sub.2] [e.sub.2] ([u.sub.1]) +...+ [W.sub.n] [e.sub.n] ([u.sub.1]).

Since f has no umbilic point on U, the Gauss map v depends only on [u.sub.1] and[mathematical expression not reproducible] holds. Therefore,

[gamma]([u.sub.1]) := v([u.sub.1] 0 ..., 0)

is a regular curve in [S.sup.n]. By a coordinate change of [u.sub.1], we may take a new coordinate system (W; t, [w.sub.2],..., [w.sub.n]) such that the spherical regular curve t [right arrow] [gamma](t) is parametrized by arc-length. Thus, we have

(3.4) f (t,[w.sub.2], ..., [w.sub.n])

= [sigma] (t)+ [w.sub.2] [e.sub.2](t) + ... + [w.sub.n] [e.sub.n](t).

Denote by e(t) := [gamma](t) the unit tangent vector of [gamma](t). Since [mathematical expression not reproducible] for each j = 2,... ,n and [gamma (t) is the Gauss map of f, we have

(3.5) [gamma](t) x [e.sub.j](t) = 0 (j = 2, ..., n).

In addition, the third equation of (3.1) yields

(3.6) [gamma] (t) x [e.sub.j](t)= 0 (j = 2, ..., n).

Therefore, [mathematical expression not reproducible] is an orthonormal frame of the normal bundle ([gamma](t)).sup.[perpendicular to]] along the spherical regular curve [gamma](t). Moreover, Eqs. (3.5) and (3.6) yield

(3.7) [gamma](t) x [e'.sub.j](t)=0 (j = 2, ..., n).

Hence, by (3.4), [f.sub.t] * [gamma] = 0 implies [sigma]'(t) x [gamma](t) = 0. Therefore, there exist smooth functions [a.sub.j] = [a.sub.j](t) (j = 1, ..., n) such that

(3.8) [sigma]'(t) = [a.sub.1](t)e(t)

+ [a.sub.2] (t) [e.sub.2] (t) + ... + [a.sub.n](t)[e.sub.n](t).

Thus, we have the following

Lemma 7. Let f : [M.sup.n] [right arrow] [R.sup.n+1] be a nontotally-umbilic flat front. For each non-umbilic point q [member of] [M.sup.n] \ [U.sub.f], there exist a local coordinate neighborhood (W; t, [w.sub.2], ..., [w.sub.n]) of q, a regular curve [gamma](t) in [S.sup.n], an orthonormal frame {[e.sub.2](t), ..., [e.sub.n](t)} of the normal bundle [([gamma]).sup.[perpendicular to]] along [gamma](t) and smooth functions [{[a.sub.j](t)}.sub.j=1] n such that f is given by (3.4) on W, where

(3.9) [mathematical expression not reproducible]

Finally, we shall reduce the numbers of functions. For a unit speed regular curve [gamma] = [gamma](t) : I [right arrow] [S.sup.n] defined on an open interval I, set e(t) := [gamma](t). Then, there exist an orthonormal frame [{[e.sub.j](t)}.sub.j=2, ... n] of the normal bundle along [gamma] and smooth functions [[mu].sub.j] (t) (j = 2, ..., n) such that

[e'.sub.j] (t) = [[mu].sub.j](t)e(t)

for each j = 2 ,..., n. Such a frame [{[e.sub.j](t)}.sub.j=2,...n] is called the Bishop frame (cf. [1]).

Let f = f (t , [w.sub.2], ..., [w.sub.n]) be the flat front given by (3.4) with the Bishop frame [{[e.sub.j](t)}.sub.j=2, ... n]. Set

p(t, [w.sub.2], ..., [w.sub.n]) := [[alpha]sub.1](t)- [N.summation over (j=2)][w.sub.j][[mu].sub.j](t).

Since [mathematical expression not reproducible]

and dv x dv = [dt.sup.2], the lift metric [ds.sup.2.sub.#] = df x df + dv dv is given by

[mathematical expression not reproducible].

By a straightforward calculation, it can be checked that each [w.sub.j]-curve (j = 2, ..., n) gives a geodesic of the lift metric [ds.sup.2.sub.#]. Thus, if f is weakly complete, every [w.sub.j]-curve (j = 2, ..., n) can be defined on the whole real line R. For each j = 2, ..., n, set

[mathematical expression not reproducible]

By a coordinate change

(t,[w.sub.2], ..., [w.sub.n]) [right arrow] (t, [w.sub.2] + [b.sub.2] (t), ..., [w.sub.n] + [b.sub.n] (t)),

we have

[mathematical expression not reproducible]

where we set [??](t) as

[mathematical expression not reproducible].

By (3.9), we have [mathematical expression not reproducible], where

a(t):= [a.sub.1](t) - [b.sub.2] (t) [[mu].sub.2] (t)-...- [b.sub.n](t)[[mu].sub.n (t).

Therefore, we have the following

Proposition 8. Let f : [M.sup.n] [right arrow] [R.sup.n+1] be a weakly complete flat front which is not totallyumbilic. Around each non-umbilic point, there exist an interval I, a local coordinate system (I x [R.sup.n-1]; t, [w.sub.2], ..., [w.sub.n]), a regular curve [gamma] : I [right arrow] [S.sup.n] parametrized by arc-length, an orthonormal frame {[e.sub.2], ..., [e.sub.n]} of the normal bundle [([[gamma].sup.l]).sup.[perpendicular to]] along [gamma] and a smooth function a(t) on I such that f is given by

(3.10) [mathematical expression not reproducible]

on I x [R.sup.n-1]. Conversely, for a given unit speed regular curve [gamma] : I [right arrow] [S.sup.n] defined on an interval I, an orthonormal frame {[e.sub.2], ..., [e.sub.n]} of the normal bundle [([[gamma].sup.l]).sup.[perpendicular to]] along [gamma] and a smooth function a(t) on I, f : I x [R.sup.n-1] ! [R.sup.n+1] defined as (3.10) is an umbilic-free flat front.

Theorem 9. Let f : [M.sup.n] [right arrow] [R.sup.n+1] be a weakly complete flat front. If n [greater than or equal to] 3 and the singular set S(f) is not empty, then S(f) cannot be compact.

Proof. Take a singular point q [member of] S(f). By Facts 3 and 5, q is not an umbilic point. By Proposition 8, we have that f is given by (3.10) on U := I x [R.sup.n-1]. Without loss of generality, {[e.sub.2], ..., [e.sub.n]} is the Bishop frame such that [e'.sub.j](t) = [[mu].sub.j](t)e(t) holds for each j = 2, ..., n. We remark that the curvature function [k.sub.[gamma]](t) of [gamma](t) is given by [k.sub.[gamma]](t) = [square root of [([[mu].sub.2](t))sup.2] + ... [([[mu].sub.n](t))sup.2]].

Differentiating (3.10), we have

[mathematical expression not reproducible]

for j = 1, ..., n, where

[mathematical expression not reproducible].

Since

[mathematical expression not reproducible]

we have

[mathematical expression not reproducible].

Let [S.sub.1], [S.sub.2] be the subsets of S(f) [intersection] U defined by

[S.sub.1] := {(t, [w.sub.2], ..., [w.sub.n]) [member of] U; a(t) = [k.sub.[gamma]](t) = 0},

[S.sub.2] := {(t, [w.sub.2], ..., [w.sub.n]) [member of] U;

[mathematical expression not reproducible],

respectively. Then, we have S(f) [intersection] U = [S.sub.1] [union] [S.sub.2].

Since v(t, [w.sub.2], ..., [w.sub.n]) = [gamma](t) gives a unit normal vector field along f, the lift metric [ds.sup.2.sub.#] is given by

[mathematical expression not reproducible]

on U.

If q = ([t.sup.o], [w.sup.o.sub.2], ..., [w.sup.o.sub.n]) [member of] [S.sub.1], a([t.sup.o]) = [k.sub.[gamma]]([t.sup.o]) (to) = 0 holds. In this case, we have ([t.sup.o], [w.sup.o.sub.2], ..., [w.sup.o.sub.n]) [member of] [S.sub.1] for any [w.sub.j] [member of] R (j = 2, ... ,n). In particular, [c.sub.1] : R [right arrow] [S.sub.1] ([subset] [M.sup.n]) given by

[c.sub.1](x) : = ([t.sup.o] , x, 0, ..., 0)

is a geodesic with respect to the lift metric [ds.sup.2.sub.#] such that [mathematical expression not reproducible] is a straight line in [R.sup.n+1], and hence S(f)([contains] [S.sub.1]) cannot be compact.

If q = ([t.sup.o], [w.sup.o.sub.2],..., [w.sup.o.sub.n]) [member of] [S.sub.2] [k.sub.[gamma]]([t.sup.o]) [not equal to] 0. Without loss of generality, we may assume that [[mu].sub.n]([t.sup.o]) [not equal to] 0. Then, there exists [epsilon] > 0 such that [[mu].sub.n]([t.sup.o]) [not equal to] 0. for each t [member of] I([t.sup.o] , [epsilon]) :=([t.sup.o] - [epsilon], [t.sup.o] + [epsilon]). Thus,

[mathematical expression not reproducible]

is a subset of [S.sub.2], where [mathematical expression not reproducible]. Set a positive number [mathematical expression not reproducible]. Since [c.sub.2] : R [right arrow] [S.sub.2][[t.sup.o]] given by

[mathematical expression not reproducible]

is a geodesic with respect to the lift metric [ds.sup.2.sub.#] such that [mathematical expression not reproducible] is a straight line in [R.sup.n+1], and hence S(f)([contains] [S.sup.2][[t.sup.o]]) cannot be compact.

Proof of Theorem 1. We shall give a proof by contradiction. Let f : [M.sup.n] [right arrow] [R.sup.n+1] a complete flat front (n [greater than or equal to] 3). By Fact 2, f is weakly complete. Assume that the singular set S(f) is not empty. By the completeness of f, the singular set S(f) must be compact, which contradicts Theorem 9. Hence, we have that S(f) must be empty, and then f is a complete flat regular hypersurface.

Acknowledgements. The author would like to thank Profs. Masaaki Umehara, Kotaro Yamada and the referee for their valuable comments. This work is supported by JSPS KAKENHI Grant Number 16K17605.

doi: 10.3792/pjaa.94.25

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Atsufumi HONDA

Department of Applied Mathematics, Faculty of Engineering, Yokohama National University, 79-5 Tokiwadai, Hodogaya-ku, Yokohama, Kanagawa 240-8501, Japan

(Communicated by Kenji FUKAYA, M.J.A., Feb. 13, 2018)
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Author:Honda, Atsufumi
Publication:Japan Academy Proceedings Series A: Mathematical Sciences
Article Type:Report
Date:Mar 1, 2018
Words:4479
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