Projections of Riemannian surfaces

Dimensions of projections of sets on Riemannian surfaces of constant curvature

Zoltán M. Balogh, Annina Iseli Mathematisches Institut, Universität Bern, Sidlerstrasse 5, CH-3012 Bern, Switzerland zoltan.balogh@math.unibe.ch annina.iseli@math.unibe.ch
Abstract.

We apply the theory of Peres and Schlag to obtain generic lower bounds for Hausdorff dimension of images of sets by orthogonal projections on simply connected two-dimensional Riemannian manifolds of constant curvature. As a conclusion we obtain appropriate versions of Marstrand’s theorem, Kaufman’s theorem and Falconer’s theorem in the above geometrical settings.

Key words and phrases:
Hausdorff dimension, Orthogonal projections
2010 Mathematics Subject Classification: 28A78
This research was partially supported by the Swiss National Science Foundation.

1. Introduction

Since orthogonal projections are Lipschitz maps, they decrease the Hausdorff dimension of sets. For example, if we take a set with then for all angles where is the orthogonal projection onto the line through the origin in which makes an with the -axis. Marstrand [12] and later Kaufman [11] proved that that there is a generic lower bound on the dimension distortion, namely that the equality holds for almost every . An improvement of these result estimating the size of exceptional sets is due to Falconer [7]. For higher dimensional generalization and a unified exposition of this type of results we refer to the books [13], [15], as well as to the expository articles [6] and [14].

It is a purpose of general interest to extend the above results to various settings of non-Euclidean geometries. In this sense we mention the recent works [1, 2, 8] for the treatment of these questions in the setting of the Heisenberg groups. Due to the complicated sub-Riemannian geometry of the Heisenberg group the above mentioned results are much weaker and much less complete than their Euclidean counterparts. It is expected that better results could be obtained in the setting of Riemannian manifolds. Various questions of geometric measure theory have been already been addressed in the setting of Riemannian manifolds. This includes the work of Brothers [4, 5]Ê in connection to Besicovitch-Federer type characterization of purely unrectifiable sets in terms of projections in the setting of homogenous spaces and also the more recent work of Hovila, Järvenpää, Järvenpää and Ledrappiar [9, 10] on two-dimensional Riemann surfaces. To our knowledge no Marstrand type result is yet available in the setting of curved geometries. The purpose of this note is a first step in this direction.

Our main result shows that on simply connected two-dimensional Riemannian manifolds of constant curvature, the same projection theorems hold as in the planar case. To formulate our main result we consider to be a two-dimensional simply connected Riemannian manifold with constant curvature and be a fixed point. If then the orthogonal projections onto geodesic lines emanating from are well defined in the whole space . Here is the geodesic line in direction i.e. the image of the line under the exponential map at . If then the orthogonal projection as above is only defined on compact sets . The main result of this note is formulated as follows:

Theorem 1.1.

Let be a complete, simply connected two-dimensional Riemannian manifold with constant curvature , a base point, and be a compact subset of . If we assume that . Denote by the orthogonal projection onto the geodesic line emanating from in direction . Then for all Borel sets the following statements hold.

  1. If , then

    1. for -a.e. .

    2. .

  2. If , then

    1. for -a.e. .

    2. For ,

Our proof is based on the theory of Peres and Schlag [16] which provides a general abstract framework of generic Hausdorff dimension distortion results in metric spaces. The statements of Threorem 1.1 will follow by the verification of the crucial conditions of regularity and transversality of projections allowing the application of the results from [16]. This is based on considerations using hyperbolic trigonometry for the case of negative curvature and spherical trigonometry in the case of positive curvature.

The structure of the paper is as follows: In the first section we recall the notation and the statement of the main result from [16] and reduce the statement of Theorem 1.1 to the hyperbolic and spherical case. In the second section we prove the statement of the main theorem in the hyperbolic case and in the third section we consider the spherical case. The last section is for final remarks.

Acknowledegments: We thank the referee for carefully reading the paper and for helpful remarks improving our presentation.

2. Preliminaries

We will now give a short summary of Peres and Schlag’s theory [16] and recall one of their main results that we will apply to the Riemannian setting in the following sections. A nice summary of Peres and Schlag’s work (inlcuding outlines of the main proofs) can also be found in [14] or [15].

Let be a compact metric space, an open interval and a continuous map

(2.1)

We think of as a family of projections over the parameter interval . Let and two distinct points. We define

(2.2)
Definition 2.1.
  1. We say that has bounded derivatives in , if: For all the function is smooth and for all compact intervals and all , there exists a constant such that for all and ,

  2. We call an interval of transversality of order for , or shorter, the transversality property is satisfied, if there exists a constant , such that for all pairs of distinct points and ,

  3. We say that is -regular, if for each there exist a constant such that for all and distinct points ,

This definition allows us to state the following theorem due to Peres and Schlag [16].

Theorem 2.2.

Let be a compact metric space which is bi-Lipschitz equivalent to a subset of a Euclidean space; an open interval and a continuous map as described in (2.1). Assume that conditions (a), (b) and (c) of Definition 2.1 are satisfied. Then the following statements hold for all Borel sets .

  1. If , then

    1. for -a.e. ,

    2. .

  2. If , then

    1. for -a.e. ,

    2. For , .

Theorem 1.1 will follow from Theorem 2.2 once we show that for orthogonal projection on the conditions from Definition 2.1 are satisfied. On the other hand, simply connected, complete two-dimensional Riemannian manifolds with constant curvature are isometric to endowed with the metric , where denotes the hyperbolic metric on for and endowed with the metric , where denotes the usual spherical metric on the for . This implies that it is enough to verify the conditions of Definition 2.1 for the cases of and .

3. Projections in

3.1. Geodesic projections in

Let denote the hyperbolic plane and the hyperbolic metric on . Let be a fixed base point in and a vector of length in the tangent plane of at . We denote by the geodesic starting at in direction and by the geodesic starting at in the direction . This defines the geodesic line through . For all angles define to be the unique vector of length in such that the counter-clockwise angle from to is . Let be the geodesic starting from in direction and be the geodesic starting from in direction . This defines the geodesic line .

For a point , let be the unique point on that minimizes the distance between and . In other words, is the unique point of that satisfies,

The existence and uniqueness of such a point holds in general for negatively curved spaces (see e.g. Proposition 2.4 in [3], page 176). This allows us to define the mapping :

Proposition 2.4 of [3] implies, that is distance non-increasing and that for each the geodesic connecting to is orthogonal to . Therefore, we will refer to the mapping as the orthogonal projection of onto .

In order to be consistent with the notion of projection used in [16] we define the generalized projection

(3.1)

where the sign ”” is to be understood as follows:

Note that it is immediate from the definition of and that

(3.2)

for all and , where denotes the Euclidean metric on . Moreover, note that is a continuous map as described in (2.1). The interval of parameters from (2.1), here is an interval of angles . The fact that , for all , is a distance non-increasing mapping, implies that is distance non-increasing, i.e., -Lipschitz, for all . In particular, this implies that the dimension of a set can not increase under the projection , .

In order to express in a way that allows us to study its transversality and regularity properties, we use basic facts from hyperbolic trigonometry. Consider a geodesic triangle in with side lengths and opposite angles . It holds that

(3.3)

This formula is called the hyperbolic law of cosines, a proof can be found for example in [3] or [17]. Applying the hyperbolic law of cosines to a right-sided triangle twice, yields

(3.4)

where . To see this, consider a triangle as just described with . From (3.3) it follows that and From these relations we obtain , which implies . Thus, (3.4) follows.

Now for each point and angle , let us denote by the counter-clockwise angle from to the geodesic segment connecting the base point to . As we will show now, (3.4) implies that

(3.5)

for all angles and all points . Let be a point in and an angle. First, we consider the case when . Then, and the three points , and span a geodesic triangle with side lengths , , and opposite angles , , . By (3.4), it follows that . Hence, by the definition of and the fact that , we obtain (3.5) for this case. The other cases: ; and can be treated similarly.

For each point , let be the counter-clockwise angle from to the geodesic segment connecting the base point to . It is easy to see that for all . In conclusion:

(3.6)

Motivated by this result, we introduce the following new family of generalized projections:

(3.7)

Note that, for all and ,

(3.8)

Thus, is a continuous mapping with respect to . Moreover, note that is -Lipschitz on the whole of . Recall, that for all , is -Lipschitz. Therefore, is -Lipschitz for all .

Now for all angles and all pairs of distinct points define,

(3.9)

analogous to (2.2) in the general setting.

3.2. Transversality and regularity properties in

Let be a compact subset of . From now on we will consider the metric space , where denotes the restriction of the hyperbolic metric to . We will consider the projections and as defined in (3.1) and (3.7), as well as the function as defined in (3.9), restricted to .

We will now show that Definition 2.1 is satisfied in this just defined setting. For this purpose, define .

Proposition 3.1.

There exist two functions

such that:

  1. For all pairs of points and all angles ,

  2. There exist constants and , such that for all ,

Proof of Proposition 3.1.

Let . Throughout this proof, we will use the following notation:

(3.10)

Moreover, we denote the counter-clockwise angle from to the geodesic segment connecting to (resp. ) by (resp. ).

By (3.5), we have and In order to make the calculations clearer, write and . Thus we obtain

(3.11)

and by an elementary calculation

(3.12)

Define

(3.13)

Note that and cannot both be , since . This allows us to make the following definition: Let be the angle that satisfies

(3.14)

In this notation, from (3.12) it follows that . Set (see below (3.10) for the definition of ) and . Observe that by their definition both and are independent of . Thus and are well-defined functions on . Moreover, by definition of and , we conclude

This completes the proof of Proposition 3.1..

For the proof of Proposition 3.1. it suffices to show that for constants and independent of and , where .

By the hyperbolic law of cosines (3.3) applied to the geodesic triangle spanned by , and , it holds that , which implies,

(3.15)

Applying (3.13) and (3.15), as well as elementary product-to-sum identities for hyperbolic and trigonometric functions, yields

(3.16)

Note that the product is greater than and is bounded from above since and is compact. So we can derive the following upper bound for :

Hence, we conclude that

Note that is a continuous function in and that Thus by the compactness of , we have for some constant only depending on the diameter of . This proves the right-hand inequality in Proposition 3.1.. Now let us prove the left-hand inequality.

Using the notation from (3.10), we define . By the triangle inequality , i.e., and therefore . The following calculation only uses the definition of and elementary calculation rules for :

From the Taylor series representation of it follows that . Consequently, the estimate,

(3.17)

follows. Now, since and compact, there exists a constant (only depending on ) such that . Thus by (3.16) and (3.17), it follows that for . This concludes the proof of Proposition 3.1. ∎

Proof of Theorem 1.1 in the negative curvature case:.

From Proposition 3.1., it follows that for all pairs of points and angle , and hence

(3.18)

Thus for all , is an element of the set

(3.19)

Consequently, from Proposition 3.1. it follows that is -regular and has bounded partial derivatives in the sense of Definition 2.1. Now let such that , where is the constant from Proposition 3.1.. Assume that . Applying Proposition 3.1, yields

and hence, . Now by (3.18), it follows that . Thus the transversality property holds as well. Now, by applying Theorem 2.2, Theorem 1.1 follows for the case when is a compact subset of . As explained in Section 2, the statement of Theorem 1.1 in the negative curvature case follows from this. ∎

4. Projections in

4.1. Geodesic projections in

Let denote the Euclidean two-sphere equipped with the usual spherical metric . Let be a fixed base point in , a fixed number and denote by the open ball of radius centered at . Let be a vector of length in the tangent plane of at . We denote by the segment of the geodesic starting at in direction that is contained in . Analogously, denote by the segment of the geodesic starting at in the direction that is contained in . This defines the geodesic segment through . For an angle let be the unique vector of length  in such that the counter-clockwise angle from to is . Let be the segment of the geodesic starting from in direction that is contained in . Analogously define in direction . This defines the geodesic segment . Note that for each direction there exists a geodesic line starting at in direction of length . So the restriction onto with might look too strong at this point. However, this restriction is crucial in order for our results to hold. We will explain this in more detail in the last section.

Let be a compact set that is contained in . Then, due to the restriction , the orthogonal projection of onto the geodesic line segment is well-defined by,

(See [3], pages 176-178.) By the same argument as in the hyperbolic plane, for a point , the geodesic segment connecting to is orthogonal to . On the other hand is not -Lipschitz. However, , for all , still is a Lipschitz map for some constant that only depends on .

Define the generalized projection , analogously to (3.1):

(4.1)

It is immediate from this definition that

(4.2)

In our considerations below we will use basic results of spherical trigonometry. The following formula is what we call the spherical law of cosines, a proof can be found for example in [3] or [17].
For a geodesic triangle with side lengths , each , and opposite angles , it holds that:

(4.3)

Applying the spherical law of cosines to a right-sided triangle twice, yields

(4.4)

where . (Note that (4.4) can be proved similarly to (3.4).) For each point , define the angle as in the hyperbolic plane (see above (3.6)). Applying an argument similar to the proof of (3.6), yields that

(4.5)

Motivated by (4.5), we define a new family of generalized projections:

(4.6)

(Compare (3.5) and (3.7).) Note that for all and ,

(4.7)

Thus, is continuous with respect to and for all , is Lipschitz, for some Lipschitz constant that only depends on .

Now for all angles and all pairs of distinct points define,

4.2. Transversality and regularity properties in

We will now show that Definition 2.1 is satisfied in the setting described in Section 4.1.

Proposition 4.1.

There exist two functions

such that:

  1. For all pairs of points and angle ,

  2. Moreover, there exist constants and , such that for all

Proof of Proposition 4.1..

Let . Throughout this proof, we will use the following notation:

(4.8)

Moreover, we denote the counter-clockwise angle from to the geodesic segment connecting to (resp. ) by (resp. ). With this notation, the proof of Proposition 4.1. is similar to the proof of Proposition 3.1..

In order to prove Proposition 4.1. it suffices to show that , for constants and independent of and . Recall that and are defined as

(4.9)

where , see (3.11) and (3.13).

By the spherical law of cosines (4.3), it holds that

Since and are both strictly smaller than , , and we obtain

(4.10)

From (4.9), (4.10) and elementary calculation rules for trigonometric functions it follows that

(4.11)

Using the fact that and thus , we can derive the following lower bound for :

This implies that

(4.12)

The function is continuous on and Since , it follows that there exists a constant , only depending on , such that . This together with (4.12) proves the left-hand inequality in Proposition 4.1..

Now let us prove the right-hand inequality. We define , thus by the triangle inequality and therefore . The following calculation only uses the definition of and elementary calculation rules for ::

Note that for . Consequently, the estimate,

(4.13)

follows. Recall that . Set , then and hence, by (4.11) and (4.13), we obtain .∎

Proof of Theorem 1.1 in the positive curvature case:.