CC BY
9
УДК 517.95
Rigidity Conditions for the Boundaries of Submanifolds in a Riemannian Manifold
Anatoly P. Kopylov* Mikhail V. KorobkoV
Sobolev Institute of Mathematics SB RAS 4 Acad. Koptyug avenue, Novosibirsk, 630090 Novosibirsk State University Pirogova, 2, Novosibirsk, 630090
Russia
Received 20.03.2016, received in revised form 28.04.2016, accepted 26.05.2016 Developing A.D. Aleksandrov's ideas, the first author proposed the following approach to study of rigidity problems for the boundary of a C0-submanifold in a smooth Riemannian manifold. Let Yi be a two-dimensional compact connected C0-submanifold with non-empty boundary in some smooth two-dimensional Riemannian manifold (X,g) without boundary. Let us consider the intrinsic metric (the infimum of the lengths of paths, connecting a pair of points".) of the interior Int Yi of Yi, and extend it by continuity (operation lim ) to the boundary points of dYi. In this paper the rigidity conditions are studied, i.e., when the constructed limiting metric defines dYi up to isometry of ambient space (X,g). We also consider the case dimYj = dimX = n, n > 2.
Keywords: Riemannian manifold, intrinsic metric, induced boundary metric, strict convexity of subman-
ifold, geodesics, rigidity conditions.
DOI: 10.17516/1997-1397-2016-9-3-320-331.
1. Introduction: unique determination of surfaces by their relative metrics on boundaries
A classical theorem says (see [3]): If two bounded closed convex surfaces in the three-dimensional Euclidean space are isometric in their intrinsic metrics then they are equal, i.e., they can be matched by a motion.
The problems of unique determination of closed convex surfaces by their intrinsic metrics goes back to the result of Cauchy, obtained already in 1813, that any closed convex polyhedrons Pi and P2 (in the three-dimensional Euclidean space) that are equally composed of congruent faces are equal. Since then this problem has been studied by many people for about 140 years (for example, by Minkowski, Hilbert, Weyl, Blaschke, Cohn-Vossen, Aleksandrov, Pogorelov and other prominent mathematicians (see, for instance, the historical overview in [3], Chapter 3); finally, its complete solution, which is just the theorem we have cited at the beginning, was obtained by A. V. Pogorelov. For generalizations of Pogorelov's result to higher dimensions, see [4].
In [5], we proposed a new approach to the problem of unique determination of surfaces, which enabled us to substantially enlarge the framework of the problem. The following model situation illustrates the essence of this approach fairly well:
* [email protected] [email protected] © Siberian Federal University. All rights reserved
Let Ui and U2 be two domains (i.e., open connected sets) in the real n-dimensional Euclidean space 1" whose closures cl Uj, where j = 1,2, are Lipschitz manifolds (such that d(cl Uj) = dUj = 0, where dE is the boundary of E in 1"). Assume also that the boundaries dUi and dU2 of these domains, which coincide with the boundaries of the manifolds cl Ui and cl U2, are isometric with respect to their relative metrics pdUj,Uj (j = 1, 2), i.e., with respect to the metrics that are the restrictions to the boundaries dUj of the extensions pci Uj (by continuity) of the intrinsic metrics pUj of the domains Uj to cl Uj. The following natural question arises: Under which additional conditions are the domains Ui and U2 themselves isometric (in the Euclidean metric)? In particular, the natural character of this problem is determined by the circumstance that the problem of unique determination of closed convex surfaces mentioned at the beginning of the article is its most important particular case. Indeed, assume that Si and S2 are two closed convex surfaces in 13, i.e., they are the boundaries of two bounded convex domains Gi C 13 and G2 C 13. Let Uj = 13 \ cl Gj be the complement of the closure cl Gj of the domain Gj, j = 1,2. Then the intrinsic metrics on the surfaces Si = dUi and S2 = dU2 coincide with the relative metrics pdUluUl and pdU2,U2 on the boundaries of the domains Ui and U2, and thus the problem of unique determination of closed convex surfaces by their intrinsic metrics is indeed a particular case of the problem of unique determination of domains by the relative metrics on their boundaries.
The generalization of the problem of the unique determination of surfaces ensuing from a new approach suggested in [5] manifests itself in the fact that the unique determination of domains by the relative metrics on their boundaries holds not only when their complements are bounded convex sets but, for example, also in the following cases.
The domain Ui is bounded and convex and the domain U2 is arbitrary (A. P. Kopylov [5]).
The domain Ui is strictly convex and the domain U2 is arbitrary (A. D. Aleksandrov (see [6])).
The domains Ui, U2 are bounded and their boundaries are smooth (V. A. Aleksandrov [6]).
The domains Ui and U2 have nonempty bounded complements, while their boundaries are (n — 1)-dimensional connected Cl-manifolds without boundary, n> 2 (V. A. Aleksandrov [7]).
In the papers [8-10], M.V. Korobkov (in particular) obtained a complete solution to the problem of unique determination of a plane (space) domain in the class of all plane (space) domains by the relative metric on its boundary.
In this connection, there appears the following question: Is it possible to construct an analog of the theory of rigidity of surfaces in Euclidean spaces in the general case of the boundaries of submanifolds in Riemannian manifolds?
Our article is devoted to a detailed discussion of this question. In it, we in particular obtain new results concerning rigidity problems for the boundaries of n-dimensional connected submanifolds with boundary in n-dimensional smooth connected Riemannian manifolds without boundary (n > 2).
In what follows, all paths 7 : [a,ft\ ^ 1", where a, ft G 1, are assumed continuous and non-constant, and l(j) means the length of a path 7.
2. Rigidity problems and intrinsic geometry of submanifolds in riemannian manifolds
Let (X, g) be an n-dimensional smooth connected Riemannian manifold without boundary and let Y be an n-dimensional compact connected C0-submanifold in X with nonempty boundary dY (n > 2).
A classical object of investigations (see, for example, [11]) is given by the intrinsic metric PdY on the hypersurface dY defined for x,y G dY as the infimum of the lengths of curves v C dY joining x and y. In the recent decades, an alternative approach arose in the rigidity theory for submanifolds of Riemannian manifolds (see, for instance, the recent articles [1,10], and [2], which
also contain a historical survey of works on the topic). In accordance with this approach, the metric on dY is induced by the intrinsic metric of the interior Int Y of the submanifold Y. Namely, suppose that Y satisfies the following condition: (i) if x,y G Y, then
PY (x,y)= liminf {inf[l(Yx',y',Int y )]} < <x>, (2.1)
x' ^x,y'^y;x',y'Glnt Y
where inf[l(Yx',y',int Y)] is the infimum of the lengths l(Yx',y',int Y) of smooth paths Yx',y',int Y : [0,1] ^ Int Y joining x' and y' in the interior Int Y of Y.
Remark 2.1. Easy examples show that if X is an n-dimensional connected smooth Riemannian manifold without boundary then an n-dimensional compact connected C0-submanifold in X with nonempty boundary may fail to satisfy condition (i). For n = 2, we have the following counterexample:
Let (X, g) be the space R2 equipped with the Euclidean metric and let Y be a closed Jordan domain in R2 whose boundary is the union of the singleton {0} consisting of the origin 0, the segment {(1 — t)(e1 + 2e2) + t(e1 + e2) : 0 < t < 1}, and of the segments of the following four types:
(1 — t)(ei + e2) + tei n n + 1
ei + (1 — t)e2 : 0 < t < U (n = 2, 3,
n
+ n+r : 0 ^ t < 1 j (n = 1, 2
(1 — t)(e1 + 2e2) 2t(2e1 + e2) . ,
(-i^-lZ—^ + ( 1 + 2) : 0 < t < U (n = 1, 2,...);
n 4n + 3 J
(1 — t)(e1 + 2e2) 2t(2e1 + e2) } ,
v-—2j_ + ^ 1 ^ 2 : 0 < t < n (n = 1, 2,...).
n +1 4n + 3 J
Here e1, e2 is the canonical basis in R2. By the construction of Y, we have pY(0,E) = to for every E G Y \ {0} (see Fig. 1).
Remark 2.2. Note that if X = R" and U is a domain in R" whose closure Y = cl U is a Lipschitz manifold (such that d(cl U) = dU = 0), then psu,u(x,y) = pY(x,y) (x,y G dU) and Y satisfies (i). Hence, this example is an important particular case of submanifolds Y in a Riemannian manifold X satisfying (i) .
To prove our rigidity results for boundaries of submanifolds in a Riemannian manifold (see Sec. 3.), we first need to study the properties of the intrinsic geometry of these submanifolds. One of the main results of this section is as follows:
Theorem 2.1. Let n = 2. Then, under condition (i), the function pY defined by (2.1) is a metric on Y.
Proof. It suffices to prove that pY satisfies the triangle inequality. Let A, O, and D be three points on the boundary of Y (note that this case is basic because the other cases are simpler). Consider e > 0 and assume that yaeo 1 : [0,1] ^ Int Y and yo2de : [2, 3] ^ Int Y are smooth paths with the endpoints A£ = YAeO1 (0), O\ = YAeO1 (1) and De = YO2De(3), Ol = YO2De(2) satisfying the conditions pX(Ae,A) < e, pX(De,D) < e, pX(Oj,O) < e (j = 1; 2), \l(YAeOie) — pY(A, O)| < e, and \1(yo¡De) — pY(O,D)| < e. Let (U,h) be a chart of the manifold X such that U is an open neighborhood of the point O in X, h(U) is the unit disk B(0,1) = {(x1, x2) G R2 : x1 + x2 < 1} in R2, and h(O) = 0 (0 = (0,0) is the origin in R2); moreover h : U ^ h(U) is a diffeomorphism having the following property: there exists a chart (Z, of Y with ^(O) = 0, A,D G U \ clx Z (clx Z is the closure of Z in the space (X,g)) and Z = U n Y is the intersection of an open neighborhood U (c U) of O in X and Y whose image ^(Z) under ^ is the half-disk B+(0,1) = {(x1,x2) G B(0,1) : x1 > 0}. Suppose that ar is an arc of the circle dB(0,r) which is a connected component of the set V n dB(0,r), where
Fig. 1. An example of 2-dimensional compact connected C0-submanifold with nonempty boundary which does not satisfy condition (i)
V = h(Z) and 0 < r < r* = min{|h(^_1(x1; x2))| : x\ + x\ = 1/4, x1 > 0}. Among these components, there is at least one (preserve the notation ar for it) whose ends belong to the sets h(^-1({-te2 : 0 < t < 1})) and h(^-1({te2 : 0 < t < 1})) respectively. Otherwise, the closure of the connected component of the set V n B(0, r) whose boundary contains the origin would contain a point belonging to the arc {eie/2 : 101 < n/2} (here we make use of the complex notation for points 2 € R2 (= C)). But this is impossible. Therefore, the above-mentioned arc
ie
z = re ar exists.
It is easy to check that if e is sufficiently small then the images of the paths h o yaoI and h°Yo2Ds also intersect the arc ar, i.e., there are ti G\0,1[, t2 G\2, 3[ such that YAeOl (ti) = xl G Z, YO2_DE (t2) = x2 G Z and h(xj) G ar, j = 1, 2. Let jr : [ti,t2\ ^ ar be a smooth parametrization of the corresponding subarc of ar, i.e., jr(tj) = h(xj), j = 1,2. Now we can define a mapping ~{e : [0, 3\ ^ Int Y by setting
f YAO (t), t G [0,ti\; 7e(t) = < h-i(Yr(t)), t G\ti,t2[;
[ YO2de (t), t G [t2,3\.
By construction, 7e is a piecewise smooth path joining the points A£ = ~/E (0), De = ~/E (3) in Int Y; moreover,
l(Ye) < Kyao )+ l(Yo2d£ ) + l(h-i(ar)).
By an appropriate choice of e > 0, we can make r > 0 arbitrarily small, and since a piecewise
smooth path can be approximated by smooth paths, we have pY(A, D) < pY(A, O) + pY(O, D).
□
In connection with Theorem 2.1, there appears a natural question: Are there analogs of this theorem for n ^ 3? The following Theorem 2.2 answers this question in the negative:
Theorem 2.2. If n ^ 3 then there exists an n-dimensional compact connected C0-manifold
Y C 1" with nonempty boundary dY such that condition (i) (where now X = 1") is fulfilled for
Y but the function pY in this condition is not a metric on Y.
Proof. It suffices to consider the case of n = 3. Suppose that A, O, D are points in 13, O is
the origin in 13, |A| = \D\ = 1, and the angle between the segments OA and OD is equal to —.
6
The manifold Y will be constructed so that O G dY, and \O,A\ C Int Y, \O,D\ C Int Y. Under these conditions, pY(O,A) = pY(O,D) = 1. However, the boundary of Y will create "obstacles" between A and D such that the length of any curve joining A and D in Int Y will be 12
greater than — (this means the violation of the triangle inequality for pY). 5
Consider a countable collection of mutually disjoint segments {j}jeN,k=1,...,kj lying in the interior of the triangle 6AAOD (which is obtained from the original triangle AAOD by dilation with coefficient 6) with the following properties:
(*) every segment j = [xk,y!j\ lies on a ray starting at the origin, yk = 11xk, and \xk\ = 2-j; (**) any curve y with ends A, D whose interior points lie in the interior of the triangle 4AAOD and belong to no segment Ik, satisfies the estimate 1(y) > 6.
The existence of such a family of segments is certain: the segments of the family must be situated chequerwise so that any curve disjoint from them be sawtooth, with the total length of its "teeth" greater than 6 (it can clearly be made greater than any prescribed positive number). However, below we exactly describe the construction.
It is easy to include the above-indicated family of segments in the boundary dY of Y. Thus, it creates a desired "obstacle" to joining A and D in the plane of AAOD. But it makes no obstacle to joining A and D in the space. The simplest way to create such a space obstacle is as follows: Rotate each segment Ik along a spiral around the axis OA. Make the number of coils so large that the length of this spiral be large and its pitch (i.e., the distance between the origin and the end of a coil) be sufficiently small. Then the set Sjj obtained as the result of the rotation of the segment Ik is diffeomorphic to a plane rectangle, and it lies in a small neighborhood of the cone of revolution with axis AO containing the segment Ij. The last circumstance guarantees that the sets Sjj are disjoint as before, and so (as above) it is easy to include them in the boundary dY but, due to the properties of the Ik's and a large number of coils of the spirals Sjj, any curve
k 12
joining A, D and disjoint from each S k has length ^ —.
We turn to an exact description of the constructions used. First describe the construction of the family of segments Ik. They are chosen on the basis of the following observation:
Let y : [0,1\ ^ 4AAOD be any curve with ends y(0) = A, y(1) = D whose interior points lie in the interior of the triangle 4AAOD. For j G N, put Rj = {xG4AAOD : \x\ G [8 • 2-j, 4 • 2-j\}. It is clear that
4AAOD \ {O} = UjeNRj.
Introduce the polar system of coordinates on the plane of the triangle AAOD with center O such
n
that the coordinates of the points A, D are r =1, y = 0 and r = 1, y = —, respectively. Given a
6
point x G 6AAOD, let yx be the angular coordinate of x in [0, 6]. Let = {yY(t) : Y(t) G Rj}. Obviously, there is jo G N such that
H^ j) > 2-jo 6, (2.2)
where H1 is the Hausdorff 1-measure. This means that, while in the layer Rj0, the curve y covers
the angular distance > 2-jo —. The segments Ik must be chosen such that (2.2) together with
6j
the condition
Y(t) n j = 0 yt G [0,1], yj G N, yk G {1,.. ., kj}
give the desired estimate 1(y) ^ 6. To this end, it suffices to take kj = [(2n)j] (the integral part of (2n)j) and
Ik = {x G 6AAOD : = k(2n)-j6, \x\ G [11 • 2-j, 2-j]} ,
k = 1,... ,kj. Indeed, under this choice of the Ik's, estimate (2.2) implies that y must intersect at least (2n)j0 2-j0 = nj0 > 3j0 of the figures
Uk = {x G Rj0 : ^x G (k(2n)-j0 6, (k + 1)(2n)-j0 6)} .
Since these figures are separated by the segments j in the layer Rj0, the curve y must be disjoint from them each time in passing from one figure to another. The number of these passages must be at least 3j0 — 1, and a fragment of y of length at least 2 • 3 • 2-j0 is required for each passage (because the ends of the segments j go beyond the boundary of the layer Rj0 containing the figures Uk at distance 3 • 2-j0). Thus, for all these passages, a section of y is spent of length at least
6 • 2-j0 (3j0 — 1) > 6.
Hence, the construction of the segments I,k satisfying (*)-(**) is finished.
Let us now describe the construction of the above-mentioned space spirals.
For x G R3, denote by nx the plane that passes through x and is perpendicular to the segment OA. On nxfc, consider the polar coordinates (p, 0) with origin at the point of intersection of nxk
and [O, A] (in this system, the point xk has coordinates p = pk, 0 = 0). Suppose that a point x(0) G nxfc moves along an Archimedean spiral, namely, the polar coordinates of the point x(0) are p(0) = pk — ej0, 0 G [0, 2nMj], where ej is a small parameter to be specified below, and Mj G N is chosen so large that the length of any curve passing between all coils of the spiral is at least 10.
Describe the choice of Mj more exactly. To this end, consider the points x(2n), x(2n(Mj — 1)), x(2nMj), which are the ends of the first, penultimate, and last coils of the spiral respectively (with x(0) = xk taken as the starting point of the spiral). Then Mj is chosen so large that the following condition hold:
(*1) The length of any curve on the plane nxk, joining the segments [xk,x(2n)] and [x(2n(Mj — 1)),x(2nMj)] and disjoint from the spiral {x(0) : 0 G [0, 2nMj]}, is at least 10.
Figuratively speaking, the constructed spiral bounds a "labyrinth", the mentioned segments are the entrance to and the exit from this labyrinth, and thus any path through the labyrinth has length > 10.
Now, start rotating the entire segment Ij in space along the above-mentioned spiral, i.e., assume that Ik(0) = {y = Xx(0) : A G [1,11]}. Thus, the segment I,(0) lies on the ray joining O with x(0) and has the same length as the original segment Ij = Ij (0). Define the surface Sk = U^e[0j2^Mj] j(0). This surface is diffeomorphic to a plane rectangle (strip). Taking e j > 0 sufficiently small, we may assume without loss of generality that 2nMjej is substantially less than pk; moreover, that the surfaces Sk are mutually disjoint (obviously, the smallness of ej does not affect property (*1) which in fact depends on Mj).
Denote by y(0) = 11x(0) the second end of the segment j (0). Consider the trapezium Pk with vertices yk, xk, x(2nM j), y(2nMj) and sides j, Ik(2nMj), [xk,x(2nMj)], and [yk, y(2nMj)] (the last two sides are parallel since they are perpendicular to the segment AO). By construction, pk lies on the plane AOD; moreover, taking ej sufficiently small, we can obtain the situation where the trapeziums Pk are mutually disjoint (since Pk ^ j under fixed M j and e j ^ 0).
Take an arbitrary triangle whose vertices lie on Pk and such that one of these vertices is also a
n , , „ „ n
Therefore, the ratio of the side of the triangle lying inside the trapezium P^ to the sum of the
vertex at an acute angle in Pk. By construction, this acute angle is at least ^ — ZAOD = .
J 2 3
3
1 n 2
other two sides (lying on the corresponding sides of Pk) is at least - sin — > -. If we consider
j 2 3 5
the same ratio for the case of a triangle with a vertex at an obtuse angle of Pk then it is greater
than 1. Thus, we have the following property:
(*2) For arbitrary triangle whose vertices lie on the trapezium Pk and one of these vertices
is also a vertex in P^, the sum of lengths of the sides situated on the corresponding sides of P^ 5
is less than ^ of the length of the third side (lying inside Pj ).
Let a point x lie inside the cone K formed by the rotation of the angle ZAOD around the ray OA. Denote by Projrotx the point of the angle ZAOD which is the image of x under this rotation. Finally, let K4aAO d stand for the corresponding truncated cone obtained by the rotation of the triangle 4AAOD, i.e., Ka^aod = {x G K : Projrotx G 4AAOD}.
The key ingredient in the proof of our theorem is the following assertion: (*3) For arbitrary space curve y of length less than 10 joining the points A and D, contained in the truncated cone K4aaOD \ {O}, and disjoint from each strip Sj, there exists a plane curve
Y contained in the triangle 4AAOD \ {O}, that joins A and D, is disjoint from all segments j
5
and such that the length of Y is less than ^ of the length of ProjrotY.
Prove (*3). Suppose that its hypotheses are fulfilled. In particular, assume that the inclusion ProjrotY C 4AAOD \ {O} holds. We need to modify ProjrotY so that the new curve be contained in the same set but be disjoint from each of the Ik's. The construction splits into several steps.
Step 1. If ProjrotY intersects a segment Ik then it necessarily intersects also at least one of the shorter sides of P k .
Recall that, by construction, Pk = ProjrotSji; moreover, y intersects no spiral strip Sk. If ProjrotY intersected Pk without intersecting its shorter sides then y would pass through all coils of the corresponding spiral. Then, by (*i), the length of the corresponding fragment of y would be > 10 in contradiction to our assumptions. Thus, the assertion of step 1 is proved.
Step 2. Denote by yp* the fragment of the plane curve ProjrotY beginning at the first point of its entrance into the trapezium Pk to the point of its exit from Pk (i.e., to its last intersection
point with Pk). Then this fragment yp* can be deformed without changing the first and the last j j
points so that the corresponding fragment of the new curve lie entirely on the union of the sides
5
of Pk; moreover, its length is less than - of the length of yp*. j 2 j
The assertion of step 2 immediately follows from the assertions of step 1 and (*2). The assertion of step 2 in turn directly implies the desired assertion (*3). The proof of (*3) is finished.
Now, we are ready to pass to the final part of the proof of Theorem 2.2.
(*4) The length of any space curve y C R3 \ {O} joining A and D and disjoint from each
strip Sk is at least —. j 5
Prove the last assertion. Without loss of generality, we may also assume that all interior points of y are inside the cone K (otherwise the initial curve can be modified without any increase of its length so that assumptions of (*4) are still fulfilled and the modified curve lies in K). If y is not included in the truncated cone K4aAOD \ {O} then ProjrotY intersects the segment [4A, 4D]; consequently, the length of y is at least 2(4 sin
ZOAD - 1) = 2(4sin 3 - l) = 2(2^ - 1) > 4, and the desired estimate is fulfilled. Similarly, if the length of y is at least 10 then the desired
estimate is fulfilled automatically, and there is nothing to prove. Hence, we may further assume without loss of generality that y is included in the truncated cone K4^AOD \ {O} and its length is less than 10. Then, by (*3), there is a plane curve 7 contained in the triangle 4AAOD \ {O},
joining the points A and D, disjoint from each segment j, and such that the length of 7 is at 5
most - of the length of ProjrotY. By property (**) of the family of segments j, the length of
12
Y is at least 6. Consequently, the length of ProjrotY is at least —, which implies the desired
5
estimate. Assertion (*4) is proved.
The just-proven property (*4) of the constructed objects implies Theorem 2.2. Indeed, since the strips Sjk are mutually disjoint and, outside every neighborhood of the origin O, there are only finitely many of these strips, it is easy to construct a C0-manifold Y C R3 that is homeomorphic to a closed ball (i.e., dY is homeomorphic to a two-dimensional sphere) and has the following properties:
(I) O G dY, [A,O[U[D,O[C IntY;
(II) for every point y G (dY) \ {O}, there exists a neighborhood U(y) such that U(y) n dY is C 1-diffeomorphic to the plane square [0,1]2;
(III) Sk C dY for all j G N,k = 1,..., kj.
The construction of Y with properties (I)-(III) can be carried out, for example, as follows: As the surface of the zeroth step, take a sphere containing O and such that A and D are inside the sphere. At the jth step, a small neighborhood of the point O of our surface is smoothly deformed so that the modified surface is still smooth, homeomorphic to a sphere, and contains all strips Sk, k = 1,... ,kj. Besides, we make sure that, at the each step, the so-obtained surface be disjoint from the half-intervals [A, O[ and [D, O[, and, as above, contain all strips Sk, i < j, already included therein. Since the neighborhood we are deforming contracts to the point O as j ^ to, the so-constructed sequence of surfaces converges (for example, in the Hausdorff metric) to a limit surface which is the boundary of a C0-manifold Y with properties (I)-(III).
Property (I) guarantees that pY(A, O) = pY(A,D) = 1 and pY(O,x) < 1 + pY(A,x) for all x G Y. Property (II) implies the estimate pY(x,y) < to for all x,y G Y \ {O}, which, granted the previous estimate, yields pY (x,y) < to for all x,y G Y. However, property (III) and the
12
assertion (*4) imply that pY(A, D) > — > 2 = pY(A, O) + pY(A, D). Theorem 2.2 is proved. □
5
If pY is a metric (the dimension n (> 2) is arbitrary) then the question of the existence of geodesics is solved in the following assertion, which implies that pY is the intrinsic metric (see, for example, §6 in [11]).
Theorem 2.3. Assume that pY is a finite function and is a metric on Y. Then any two points x,y G Y can be joined in Y by a shortest curve y : [0, L] ^ Y in the metric pY; i.e., y(0) = x, y(L) = y, and
py(Y(s),Y(t))= t — s, ys,t G [0,L], s<t. (2.3)
Proof. Fix a pair of distinct points x,y G Y and put L = pY (x,y). Now, take a sequence of paths Yj : [0, L] ^ Int Y such that Yj (0) = xj, Yj(L) = yj, xj ^ x, yj ^ y, and l(Yj) ^ L as j ^ to. Without loss of generality, we may also assume that the parametrizations of the curves yj are their natural parametrizations up to a factor (tending to 1) and the mappings yj converge uniformly to a mapping y : [0, L] ^ Y with y(0) = x, y(L) = y. By these assumptions,
lim l(Yj\[si])= t — s ys,t G [0,L], s<t. (2.4)
j^^
Take an arbitrary pair of numbers s,t G [0, L], s < t. By construction, we have the convergence Yj(s) G Int Y ^ y(s), Yj(t) G Int Y ^ Y(t) as j ^ to. From here and the definition of the
metric pY(•, •) it follows that pY(Y(s),Y(t)) ^ limj^TO l(Yj\[s,t]). By (2.4),
py(Y(s),Y(t)) < t — s ys,t G [0,L], s<t. (2.5)
Prove that (2.5) is indeed an equality. Assume that pY(y(s'), Y(t')) <t' — s' for some s ', t' G [0, L], s < t . Then, applying the triangle inequality and then (2.5), we infer
pY(x, y) < py(x, y(s')) + py(y(s'), Y(t')) + pY(Y(t'), y) <s' + (t' — s') + (L — t') = L,
which contradicts the initial equality pY(x, y) = L. The so-obtained contradiction completes the proof of identity (2.3). □
Remark 2.3. Identity (2.3) means that the curve of Theorem 2.3 is a geodesic in the metric pY , i.e., the length of its fragment between points y(s), Y(t) calculated in pY is equal to pY(y(s), Y(t)) = t — s. Nevertheless, if we compute the length of the above-mentioned fragment of the curve in the initial Riemannian metric then this length need not coincide with t — s; only the easily verifiable estimate 1(y\[s,i]) ^ t — s holds (see (2.4)). In the general case, the equality 1(y\[s,t]) = t — s can only be guaranteed if n = 2 (if n > 3 then the corresponding counterexample is constructed by analogy with the counterexample in the proof of Theorem 2.2, see above). In particular, though, by Theorem 2.3, the metric pY is always intrinsic in the sense of the definitions in [11, §6], the space (Y,pY) may fail to be a space with intrinsic metric in the sense of [ibid].
3. Rigidity theorems for the boundaries of submanifolds in Riemannian manifolds
As in Sec. 2., let (X, g) be an n-dimensional smooth connected Riemannian manifold without boundary and let px be its intrinsic metric (i.e., let px (x,y) be the infimum of the lengths l(Yx,y,x) of smooth paths Yx,y,X : [0,1] ^ X joining points x and y in a manifold X).
Assume that Y is an n-dimensional compact connected C0-submanifold Y C X with nonempty boundary dY satisfying condition (i) in Sec. 2., moreover, pY is a metric on Y. Then Y is called strictly convex in the metric pY if, for any a, 3 G Y, any shortest path Y = Ya,p,Y : [0,1] ^ Y between a and 3 (in the metric pY) satisfies y(]0,1[) C Int Y.
Theorem 3.1. Let n = 2. Assume that condition (i) holds for a 2-dimensional compact connected C0-submanifold Y1 with nonempty boundary dY1 of a 2-dimensional smooth connected Riemannian manifold X without boundary which is strictly convex in the metric pYl. Suppose that Y2 C X is also a 2-dimensional compact connected C0-submanifold of X with dY2 = 0 satisfying (i); moreover, dY1 and dY2 are isometric in the metrics pY., for j = 1, 2. Then, Y2 is strictly convex with respect to pY2.
Proof. Suppose that, for points x,y G Y2, there exists a shortest path yx,y,Y2 : [0,1] ^ Y2 in the metric pY2 joining x and y and such that {yx,y,Y2(]0,1[)} n dY2 = 0, i.e., x' = yx,y,Y2 (t') G {Yx,y,Y2 (]0,1[)ndY2} for a point t' g]0, 1[. By Theorem 2.3 and the fact that Y2 is a 2-dimensional compact connected C0-submanifold in X, for a sufficiently small neighborhood of x' in Y2, we can find points x0,y0 G dY2 and a shortest path Yx0,y0,Y2 : [0,1] ^ Y2 between x0 and y0 in the same metric satisfying the condition x' G {Yx0,y0,Y2 (]0,1[) n dY2}. Further, we will suppose that x = x0 and y = y0.
Now, assume that f : dY1 ^ dY2 is an isometry of dY1 and dY2 in the metrics pYl and pY2 of the boundaries dY1 and dY2 of the submanifolds Y1 and Y2 of X. From Theorem 2.3, we have
pY2 (x, x') + pY2 (x' , y) = l1 + l2 = l = py2 (x, y).
Since f is an isometry,
PYl(f-1(x),f-1(x'))+ PYl(f-1(x'),f-1(y)) = PY2(x, x) + PY2 (x,y)■
Next, consider shortest paths Yf-i(x),f-i{x'),y1 : [0,1/2] ^ Y1 and Yf-i(x'),f-i(y),Yl : [1/2,1] ^ Y1 in pYl between (respectively) f-1(x) and f-1(x') and f-1(x') and f-1(y), and then construct a path y : [0,1] ^ Y1 by setting Y(t) = Yf-1 (x),f-i{x'),Yl (t) if 0 < t < 1/2 and = Yf-i(x'),f-i(y),Yl (t) for 1/2 < t < 1. Let lYl (S) be the length of a path S : [0,1] ^ Y1 in the metric pYl. Since pYl is a metric on Y1, it is not difficult to show that
lYi (Y) < lYi (Yf-1(x),f-1(x'),Yi ) + lYi (Yf-1(x'),f-1(y),Yi ) = l1 + l2 ■
Hence y is a shortest path in pYl joining f-1(x) and f-1 (y) in Y1. This contradicts the strict convexity of Y1. The theorem is proved. □
Corollary 3.1. Suppose that the conditions of Theorem 3.1 hold and the manifold X has the following property: px(x,y) = pY(x,y) for any two points x and y from every 2-dimensional compact connected C0-submanifold Y C X with dY = 0 satisfying condition (i) and strictly convex with respect to the metric pY. Then, dY1 and dY2 are isometric in the metric px on the ambient manifold X.
Remark 3.1. The condition imposed on the manifold X in Corollary 3.1 can be reformulated as follows: in this manifold, every 2-dimensional compact connected C0-submanifold Y with boundary satisfying condition (i) and strictly convex with respect to its intrinsic metric pY is a convex set in the ambient space X with respect to the metric pX (for the notion of a convex set in a metric space the reader is referred, for example, to [11]).
We have the following analog of Theorem 3.1 and Corollary 3.1 (combined together) for n ^ 3:
Theorem 3.2. Let n ^ 3. Suppose that (X,g) is an n-dimensional smooth connected Rie-mannian manifold without boundary and Y1 and Y2 are n-dimensional compact connected C0-submanifolds with nonempty boundaries dY1 and dY2 in X satisfying conditions (i),
(ii) pYj is a metric on Yj (j = 1, 2), and
(iii) for any two points a,b G Yj, there exist points c,d G dYj which can be joined in Yj by a shortest path y : [0,1] ^ Yj in the metric pY. so that a,b G y([0, 1]).
Furthermore, assume that Y1 is strictly convex in the metric pYl, X has the additional property that pX (x,y) = pY (x,y) for any two points x and y in every n-dimensional compact connected C0-submanifold Y C X with dY = 0 satisfying conditions (i) —(iii) and strictly convex with respect to pY and the boundaries dY1 and dY2 of the submanifolds Y1 and Y2 are isometric with respect to the metrics pYj, where j = 1, 2. Then, dY1 and dY2 are isometric with respect to px.
Remark 3.2. For a submanifold Y in X, (i) and (ii) can be considered as conditions of generalized regularity near its boundary.
Remark 3.3. Theorem 3.1, Corollary 3.1, and Theorem 3.2 are closely related to a theorem of A. D. Aleksandrov about the rigidity of the boundary dU of a strictly convex domain U in Euclidean n-space 1" by the relative metric pdU,u on the boundary. The following is an important particular case of this theorem:
Theorem 3.3 (A. D. Aleksandrov ( [6])). Let U1 be a strictly convex domain in 1" (i.e., for any a, ¡3 G cl U1 every shortest path y = Ya,p,el Ul : [0,1] ^ cl U1 between a and /3 (in the metric pd u1) satisfies y(]0, 1[) C U1). Assume that U2 C 1" is any domain whose closure is a Lipschitz manifold (such that d(cl U2) = dU2 = 0); moreover, dU1 and dU2 are isometric in their relative metrics pdu1,u1 and pdu2,u2. Then dU1 and dU2 are isometric in the Euclidean metric.
We say that an n-dimensional compact (closed) connected C0-submanifold Y with boundary dY = 0 of an n-dimensional smooth connected (respectively, n-dimensional smooth complete connected) Riemannian manifold X without boundary has property (o) if jx,y,Y(]0,1[) С Int Y for any two points x,y G dY and for every shortest path jx,y,Y : [0,1] ^ Y in the metric pY joining these points.
Theorem 3.4. Let n = 2. Suppose that (i) holds for a 2-dimensional compact connected C0-submanifold Y\ with boundary dYi = 0 in a 2-dimensional smooth connected Riemannian manifold X without boundary; moreover, Y\ has property (o). Assume that Y2 С X is a 2-dimensional compact connected C0-submanifold with dY2 = 0 in X and dYi and dY2 are isometric in the metrics pYj (j = 1, 2). Then dY2 also has property (o).
This theorem has the following generalization.
Theorem 3.5. Let n = 2. Suppose that (i) holds for a 2-dimensional closed connected C0-submanifold Y\ with boundary dYi (= 0) in a 2-dimensional smooth complete connected Riemannian manifold X without boundary satisfying (o). Assume that Y2 С X is a 2-dimensional closed connected C0-submanifold with dY2 = 0 in X; moreover, dYi and dY2 are isometric in the metrics pYj (j = 1, 2). Then Y2 has the property (o) as well.
Corollary 3.2 (of Theorem 3.4). Assume that the hypothesis of Theorem 3.4 hold and that the manifold X has the following property: pX (x,y) = pY (x,y) for any two points x and y on the boundary dY of every 2-dimensional compact connected C0-submanifold Y С X with dY = 0 satisfying (i) and (o). Then dYi and dY2 are isometric in the metric pX of the ambient manifold X.
Corollary 3.3 (of Theorem 3.5). Assume that the hypothesis of Theorem 3.5 hold and that the manifold X has the following property: pX (x,y) = pY(x,y) for any two points x and y on the boundary dY of every 2-dimensional closed connected C0-submanifold Y С X with dY = 0 satisfying (i) and (o). Then dYi and dY2 are isometric with respect to pX.
Theorem 3.6. Let n ^ 3. Suppose that (X,g) is an n-dimensional smooth connected Rie-mannian manifold whithout boundary and Yi and Y2 are n-dimensional compact connected C0-submanifolds with nonempty boundaries dYi and dY2 in X satisfying conditions (i) and (ii) (in Theorem 3.2). Assume that Yi has property (o) and X satisfies the following condition: pX (x,y) = pY (x,y) for any two points x and y on the boundary dY of every n-dimensional compact connected C0-submanifold Y С X with dY = 0 satisfying (i), (ii), and (o). Suppose also that dYi and dY2 are isometric in the metrics pYj, where j = 1, 2. Then dYi and dY2 are isometric in pX.
Theorem 3.7. Let n ^ 3. Suppose that (X,g) is an n-dimensional smooth complete connected Riemannian manifold without boundary and Yi and Y2 are n-dimensional closed connected C0-submanifolds with nonempty boundaries dYi and dY2 in X satisfying (i) and (ii). Assume that dYi has property (o) and X satisfies the following condition: pX (x,y) = pY(x,y) for any two points x and y on the boundary dY of every n-dimensional closed connected C0-submanifold Y with dY = 0 in X satisfying (i), (ii), and (o). Suppose also that dYi and dY2 are isometric in the metrics pYj (j = 1, 2). Then dYi and dY2 are isometric in pX.
Proofs of Theorems 3.2 and 3.4-3.7 are similar to the proof of Theorem 3.1 (Theorems 3.2 and 3.4-3.7 can be proved using the corresponding analogs of Theorems 2.1 and 2.3).
In conclusion, note that main results of our article were earlier announced in [1] and [2].
The authors were partially supported by the RFBR for, grants 14-01-00768-a and 15-01-08275-a.
References
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Об условия жесткости границ подмногообразий риманового многообразия
Анатолий П. Копылов Михаил В. Коробков
Институт математики им. С. Л. Соболева СО РАН пр. ак. Коптюга, 4, Новосибирск, 630090 Пирогова, 2, Новосибирск, 630090
Россия
В процессе развития идей академика А. Д. Александрова первым автором был предложен следующий подход к изучению проблем жесткости для краёв C0-подмногообразий в некотором гладком римановом многообразии. Пусть Yi представляет собой двумерное компактное связное C0-подмногообразие с непустым краем в некотором гладком двумерном римановом многообразии (X,g) без края. Рассмотрим внутреннюю метрику (инфимум длин путей, соединяющих данную пару точек) внутренности Int многообразия Y\ и продолжим ее по непрерывности (операцией lim) на краевые точки dYi. В настоящей статье 'рассматривается вопрос о жесткости, т.е. когда указанная метрика определяет dYi с точностью до изометрии в объемлющем пространстве (X,g). Рассматривается также случай dim Yj = dim X = n, n > 2.
Ключевые слова: риманово многообразие, внутренняя метрика, индуцированная метрика на крае, строгая выпуклость многообразия, геодезические, условия жесткости.
