Research Article | Open Access

Izu Vaisman, "Hessian Geometry on Lagrange Spaces", *International Journal of Mathematics and Mathematical Sciences*, vol. 2014, Article ID 793473, 9 pages, 2014. https://doi.org/10.1155/2014/793473

# Hessian Geometry on Lagrange Spaces

**Academic Editor:**Aloys Krieg

#### Abstract

We extend the correspondence between Hessian and Kähler metrics and curvatures to Lagrange spaces.

#### 1. Introduction

Hessian geometry on locally affine manifolds was studied by several authors, particularly, Cheng and Yau [1] and Shima [2]. Shima introduced a notion of Hessian curvature, which is a finer invariant than Riemannian curvature (the Riemannian curvature of metrics defined by the Hessian of a function was studied extensively, for example, [3]) and is related to the curvature of an associated Kähler metric on the tangent manifold (the total space of the tangent bundle). A Lagrange space is a manifold with a regular Lagrangian, also called a Lagrange metric, on its tangent manifold [4]. The latter has the vertical foliation by fibers and the fiber-wise Hessian of the Lagrangian defines (pseudo)Hessian (“pseudo” is added if the metric is not positive definite) metrics of the fibers. In this note, we extend the correspondence Hessian versus Kähler to the vertical foliation of the tangent manifold of a Lagrange space (Section 3). The subject of the note is not Lagrangian dynamics but Hessian geometry and curvature in the context of Lagrange spaces, which are a generalization of (pseudo)Finsler spaces. The study of curvature is motivated by the general principle that curvature invariants differentiate between spaces of a given type. We will begin by recalling the basics of Hessian and tangent bundle geometry (Section 2) (since the reader is not supposed to be an expert on any of these) and by some required preparations. In an appendix we give index-free proofs of some properties of Hessian curvature established via local coordinates in [2]. We work in the category and use the standard notation of differential geometry [5].

#### 2. Preliminaries

This is a preliminary section where we recall Hessian metrics and curvature and the basics of the geometry of tangent bundles. We refer to [2] for Hessian geometry and to [4, 6] for the tangent bundle geometry.

##### 2.1. Hessian Geometry

Let be a locally affine manifold with the flat, torsionless connection . A* (pseudo)Hessian metric (structure)* on is a (pseudo)Riemannian metric such that
where is an open covering of , are local, parallel vector fields, and . If (1) holds on with a function , the metric is* globally (pseudo)Hessian*. Since local parallel vector fields are of the form (in the paper we use the Einstein summation convention), where are local affine coordinates and ., (1) is equivalent to

Let be an arbitrary (pseudo)Riemannian metric on . The formula
defines a tensor, which we call the* Cartan tensor*. If the arguments are parallel vector fields, in particular vectors , the result is
The latest formula shows that is a (pseudo)Hessian metric with components as in (2) if and only if the tensor is totally symmetric.

The following question is natural: what are the conditions that characterize the class of (pseudo)Hessian manifolds within the class of (pseudo)Riemannian manifolds ? The most straightforward answer (a significant answer to the question was given in [7]) is that a (pseudo)Riemannian manifold is (pseudo)Hessian if and only if the Levi-Civita connection of can be deformed into a torsion-less flat connection and the Cartan tensor of the resulting pair is symmetric.

This remark motivates the introduction of the* difference (deformation) tensor * [2], which has the following obvious properties:
where is parallel in the first equality. The second equality is a consequence of the first since two parallel vector fields commute and has no torsion. The following lemma computes the difference tensor in the (pseudo)Hessian case.

Lemma 1. *If the metric is (pseudo)Hessian, then
*

*Proof. *Since is a tensor, it suffices to evaluate on parallel vector fields, which we shall assume for all the arguments below; hence, (4) holds. On the other hand, from the well-known global expression of the Levi-Civita connection (see volumes I and IV.2 of [5]) and since the bracket of two parallel vector fields vanishes, we have
If is (pseudo)Hessian, is symmetric and we get the required result.

Condition requires the relation between the curvatures of , . For arbitrary connections, a technical calculation that starts with the definitions gives the known formulas:
where the arguments are arbitrary tangent vectors of , denotes curvature, denotes torsion and
is a* mixed covariant derivative*. As a consequence, we get the following necessary condition for a (pseudo)Hessian metric.

Proposition 2. *If is a (pseudo)Hessian metric, there exists a symmetric deformation tensor such that the Riemannian curvature of satisfies the relation
**
where
*

*Proof. *Use and use parallel arguments in (8).

We shall return to the terminology of [2] as follows.

*Definition 3. *The* Hessian curvature operator and tensor* are, respectively, given by
(in the covariant curvature tensor we have preferred an order of arguments which is different from the order used in [2]).

Proposition 4. *The Hessian curvature tensor of a (pseudo)Hessian metric is related to the mixed covariant derivative by the relations
**
where the arguments are parallel vector fields.*

*Proof. *The difference is provided by (9) and (12). Then, using the first equality (5), (6), the symmetry of , and the preservation of by , we get the required formula.

Formulas (13) lead to the following reformulation of the necessary condition given in Proposition 2 for (pseudo)Hessian metrics.

Proposition 5 (see [2]). *If is a (pseudo)Hessian metric, the following relation between the Riemannian and the Hessian curvature of holds:
*

*Proof. *We may assume that the arguments are parallel vector fields. Then, (10) and (13) imply
(we have used the commutation of parallel vector fields and the properties and .) This is equivalent to (14).

*Remark 6. *We refer the reader to [2] for the expression of the local components of the Hessian curvature. See also the appendix of the present paper for the evaluation of the Hessian curvature on parallel arguments.

##### 2.2. Tangent Bundle Geometry

Let be an -dimensional manifold and its tangent bundle with the total space , called the* tangent manifold* of . The differentiable structure of is given by local coordinates , where , , , are local coordinates on and are vector coordinates with respect to the basis . The corresponding coordinate transformations are
The fibers of define the* vertical foliation *. We will use the same symbol for the tangent bundle of the vertical leaves.

A tangent vector has a* vertical lift* defined by
The formula (, ) defines a Nijenhuis tensor field (a Nijenhuis tensor field is an endomorphism of that has a vanishing Nijenhuis tensor , , , , , ) with the properties , , called the* tangent structure* of .

A* tangent metric* on is a (pseudo)Riemannian metric with a nondegenerate restriction to and such that
Then, is a complement of in , is a nondegenerate metric on , and defines an isometry with the inverse , which we extend by on . The mapping is a bijection between tangent metrics and pairs consisting of a complementary bundle and a transversal metric of the foliation . If we start with the pair , is defined by and

Any complement of () is called a* horizontal bundle* and also a* nonlinear connection*. A vector has a* horizontal lift * characterized by . The horizontal lifts of yield local tangent bases of ,
with the dual cotangent bases , where are local functions on , known as the coefficients of the nonlinear connection . If a horizontal bundle was chosen, it is convenient to look at the transversal tensors of (*horizontal tensors*) as tensors on by extending them by zero if evaluated on at least one vertical argument and, similarly, to extend -tensors (*vertical tensors*) by zero on a horizontal argument. On the other hand, we can* reflect* vertical tensors to horizontal tensors and vice versa by first applying , , respectively, to the arguments. The reflection of will be denoted by , if is horizontal and it will be denoted by , if is vertical.

Let us fix a decomposition . A* Bott connection* is a linear connection on that preserves the subbundles , and satisfies the conditions
where “pr” stands for “projection.”

On there exists a unique Bott connection that preserves the tensor fields , ; it is given by adding to (21) the derivatives
for all , , , . is called the* Berwald connection* [4].

If is an arbitrary linear connection, we get an* associated Bott connection *(called a* Vrănceanu connection* in [8], where the author traced back the history of this connection to a 1931 paper by Vrănceanu [9]) given by (21) and

If is a tangent metric such that , the Bott connection associated with the Levi-Civita connection of will be called the* canonical connection* of . It is the unique Bott connection that satisfies the conditions [10, 11]
The restriction of the canonical connection to the vertical leaves is the Levi-Civita connection of the restriction of to the leaves. We refer the reader to [6] for the curvature properties of the canonical connection.

*Remark 7. *The definition of Bott and canonical connection extends to arbitrary foliations on a (pseudo)Riemannian manifold and the characterization (24) is correct in the general case [10, 11].

Formula (3) with replaced by and with vertical arguments yields a vertical Cartan tensor associated with the tangent metric . In tangent bundle geometry, usually, it is the horizontal reflection that is called the (horizontal) Cartan tensor. A local calculation that uses the bases (20) yields the formula where the arguments are horizontal and is the pair associated with .

A tangent metric is called a* Lagrange metric* if the corresponding tensor is given by , where the (continuous and smooth outside the zero section) function on is a* regular Lagrangian* (regularity means that is nondegenerate). The pair is called a* Lagrange manifold* [4].* Finsler metrics* are Lagrange metrics with a Lagrangian of the form , where is positive and positive homogeneous of degree and the corresponding Lagrange metric is positive definite [12]. Then, is a* Finsler manifold*. The functions are also called a Lagrange and Finsler metric, respectively.

The tensor is totally symmetric if and only if is a* locally Lagrange metric*; that is, each point has a neighborhood, where is a Lagrange metric. But, such a metric is a globally Lagrange metric if and only if some cohomological obstructions vanish [6].

*Remark 8. *A regular Lagrangian defines a canonical horizontal bundle called the* Cartan nonlinear connection* [4, 12] and there exists a canonical tangent metric associated with the pair . The restriction of the canonical connection of a tangent metric to may be called the* Chern connection*, because it coincides with the* Rund-Chern connection* in the case of Finsler manifold. This is shown by a comparison of the connection coefficients given by formula of [10] and formula of [12]. The -reflection of the restriction of the canonical connection of to will be called the* Hashiguchi connection*, again, because it yields the connection bearing this name in Finsler geometry, as shown by a comparison of the connection coefficients calculated by formulas and of [10] and Theorem of [4]. Furthermore, in the Lagrange and Finsler case, the restriction of the Berwald connection to is the Berwald connection of Finsler and Lagrange geometry.

#### 3. Lagrange-Hessian Geometry

Formula (16) shows that the vertical leaves of a tangent manifold are affine manifolds with affine coordinates and a (locally) Lagrange metric produces (pseudo)Hessian metrics of the vertical leaves, which differentiably depend on the “parameters” . We refer to the geometry of this* leaf-wise (pseudo)Hessian metric* as* Lagrange-Hessian geometry*.

In particular, we will consider the* Lagrange-Hessian curvature* as follows. A vector field is parallel on the leaves if and only if is projectable to . It follows that the second formula (22) is equivalent to the fact that for vertical, parallel vector fields , and we see that is flat along the vertical leaves. is also torsionless since for any vertical, parallel vector fields , . Hence, is the connection with the role of of Section 2. On the other hand, if is a tangent metric of with the corresponding horizontal bundle and the corresponding transversal metric , the Levi-Civita connection of the vertical leaves is the restriction of the canonical connection of . Therefore, plays the role of the connection of Section 2 and we have a* difference tensor*
In particular, , , if , and properties (5) with replaced by hold.

Then, formulas (12) with vertical, parallel arguments define the notion of Lagrange-Hessian curvature of a tangent, in particular, a Lagrange, metric.

*Example 9. *Assume that the tangent metric is projectable; that is, the horizontal, tensorial components of depend only on . This implies that we are in the Lagrange case, namely, and that the Cartan tensor vanishes. Since are constant along the vertical leaves, the Christoffel symbols of each leaf vanish and formulas (13), (9), and (4) (which hold in the present case too since are just “parameters”) imply the vanishing of the Lagrange-Hessian curvature.

*Remark 10. *The Lagrange-Hessian curvature defines the Hashiguchi curvature operator in vertical directions by
where with and without indices are horizontal vectors and the upper index denotes reflection. Indeed, the definition of the Hashiguchi connection implies
whence,
Thus, formula (14) implies the required relation.

*Remark 11. *For horizontal arguments, assumed to be projectable vector fields, the torsion terms of the second formula (8) for , vanish and the curvature is

Now, we address the subject of the correspondence Hessian versus Kähler and we will show how to extend the correspondence between a Hessian metric on the locally affine manifold and a Kähler metric on the tangent manifold [2] to Lagrange-Hessian metrics of a Lagrange space . There is no need to recall the original construction of [2] because it amounts to the case of an isolated leaf in the general construction.

We consider the total space of the tangent bundle of the vertical leaves. This is a -dimensional manifold, which we will call the* vertical tangent manifold* of . The iterated tangent manifold has local coordinates , where , change by formulas (16) and , are vector coordinates with respect to the bases and change as follows:
The vertical tangent manifold is the submanifold of defined by . On the other hand, may be identified with the submanifold defined by , which is the zero section of the projection .

Formulas (31) show that the projection given by is the Whitney sum , where . will be called the* double vertical bundle (foliation)* and we will denote the identification of with the two terms of . Notice also the* flip involution * defined by .

Another way of looking at the manifold is to identify it with the total space of the complexified tangent bundle such that is the real part and is the imaginary part of the complexification. This interpretation shows that a horizontal bundle on may also be seen as a horizontal bundle on ; that is, .

Locally, on , we have tangent bases , where is given by (20) and dual bases with the same coefficients .

We recall that a CR structure is a complex tangent distribution that is involutive and such that (the bar denotes complex conjugation). On the other hand, a tangent bundle endomorphism such that is an structure. Then, has the eigenvalues and, if the -eigenbundle is involutive, is a CRF structure [13].

Proposition 12. *For any choice of a horizontal bundle, there exists a canonical CRF structure on the vertical tangent manifold .*

*Proof. *On , multiplication by defines a complex bundle structure, which provides a complex structure along the* double vertical leaves* with the local expression
The local expression shows the integrability of along the leaves. We get the required tensor by putting , .

The leaf-wise complex structure may also be defined by means of the endomorphisms defined on by the formula , where and is the tangent structure of . The tensor fields are Nijenhuis tensors of a constant rank such that . The vanishing of follows from the local expressions The structure is determined by the equalities

We shall need metrics that are the analog of tangent metrics and it is convenient to define them using the tensors .

*Definition 13. *A (pseudo)Riemannian metric on will be a* double tangent metric* if , is nondegenerate and

With the cotangent bases (32), a double tangent metric may be written as
If is a double tangent (pseudo)Riemannian metric, and are nondegenerate and is an isometry. For a function on , the horizontal tensor is still invariant and nondegenerate in the* regular* case. Accordingly, we may extend the notions of* locally Lagrange*,* Lagrange,* and* Finsler* to double tangent metrics.

It follows easily that any double tangent metric is compatible with the CRF structure tensor in the sense that
and, with the terminology of [13], is a* metric CRF structure*. This implies that the restriction of to the leaves of is Hermitian for the complex structure .

Clearly, the double tangent metrics are in a bijective correspondence with pairs , where is a horizontal bundle on and is a nondegenerate metric on . A tangent metric on defines a horizontal bundle endowed with a metric and the interpretation of by means of allows the identification of with a similar pair on . Accordingly, we get a double tangent metric on called the* extension of *. If the two first terms of (37) express the tangent metric on , (37) is the extension of the former to .

The next proposition shows the correspondence between locally Lagrange metrics on and the (pseudo)Kähler metrics on .

Proposition 14. *Let be a tangent metric on and its extension to . Then, the restriction of to the leaves of is a (pseudo)Kähler metric if and only if is locally Lagrange.*

*Proof. *Formula (33) shows that are holomorphic coordinates along the leaves . Then, from (37), we see that the metric induced by on these leaves is given by
and the corresponding Kähler form is
where are the horizontal components of the given metric . Since , it follows that along the leaves . if and only if , that is, if and only if the Cartan tensor of is symmetric; therefore, is a locally Lagrange metric.

In order to get a corresponding relationship between the Lagrange-Hessian and Kähler-Riemannian curvatures, we need an adequate connection, which is provided by the following proposition.

Proposition 15. *Let be a double tangent metric on . Then, there exists a unique Bott connection , with respect to the foliation , which has the following properties:*(1), ,(2),(3),(4),* where , , , , , .*

*Proof. *Let us extend the field of scalars to and define the Hermitian metric on by
where , are real vectors. We also extend connections to complex vector fields by requiring complex linearity. Being a Bott connection, the required preserves , and

By property , also preserves the eigenbundles of and, if , , property yields
The covariant derivatives , can be obtained from the first condition and condition like in the well-known case of a Riemannian connection [5, Proposition ]. Finally, in order to get the covariant derivatives , where , belong both either to the or the -eigenbundle, we notice that the second condition is equivalent to
where bar denotes complex conjugation. If all the arguments belong to either or , we already have the covariant derivatives and the Equality (44) determines . The obtained results also show that is the complexification of a real connection.

We will say that is the* Hermitian connection* of since, along the leaves of , is the Hermitian connection of the leaves ([5, Proposition ] and [11, Theorem ]). If the metric is the extension of a locally Lagrange, tangent metric of , then, by Proposition 14, restricts to Kähler metrics on the leaves of and the Hermitian connection of the leaves coincides with the Riemannian connection ([5, Theorem ]). This implies that the Hermitian connection satisfies the properties (24) with the foliation replaced by . Hence, is the canonical connection of the pair (see Remark 7), being the Levi-Civita connection of . This observation leads to the following result.

Proposition 16. *Let be the extension of the locally Lagrange, tangent metric of and the canonical connection of ; then, along seen as a submanifold of , one has
*

*Proof. *The submanifold has the local equations . As previously noticed, we have . Since is a real connection that commutes with , the isomorphism of complex vector bundles shows that must preserve the real part of the complexification . On the other hand, the connection induced by in the subbundle also satisfies (24); therefore, it just is the canonical connection on . Accordingly, the covariant derivatives in the two curvature tensors of (45) are the same and we are done.

Furthermore, we define the* extended Berwald connection * on to be the Bott connection with respect to the foliation and the horizontal bundle such that
for all , for all , and all parallel fields (i.e., ), where is defined by on and by zero on and . The other covariant derivatives are provided by the Bott condition (21) with , replaced by , , respectively. In particular, using a projectable field,
we see that preserves , , separately. The second part of (46) shows that the same is true for . Furthermore, using the definition of and the local formulas (33), we see that . By comparing the definitions, we also see that induces the Berwald connection on the submanifold .

Finally, we can prove the following proposition which is the announced relation between curvatures.

Proposition 17. *Let be the extension of the locally Lagrange, tangent metric of . Then, at any point of the submanifold and for any arguments , the following relation holds:
*

*Proof. *In the interpretation of as the total space of the complexified tangent bundle , and defined by taking complex arguments in formula (12) are the extension of , , , to complex arguments by -linearity. Accordingly, if transposed to complex arguments, the proof of formula (14) holds, which means that we have
where the arguments are parallel vector fields in . A known property of the curvature tensor of a Kähler metric tells us that we have
Now, for parallel arguments , (49) becomes
where the last term vanishes because it is a scalar product of orthogonal vectors. For the first term, the interpretation of as the complexification of yields

Combining the results we get the required conclusion.

If we take the arguments of (48) in the basis and decompose them into the sum of the holomorphic and antiholomorphic part, then, using the properties of the curvature tensor of a Kähler metric, we will get Proposition 3.3 of [2].

#### Appendix

In this appendix we give an index-free presentation of some more facts concerning Hessian curvature on locally affine manifolds that were treated via local coordinates in [2]. The notation is the same as in Section 2.2.

The importance of the symmetry properties of the Riemannian curvature tensor suggests looking for symmetry properties of the Hessian curvature. These follow from the following result that is equivalent to formula (1) of Proposition 3.1 of [2].

Proposition A.1. *The value of the Hessian curvature tensor on parallel arguments is given by
*

*Proof. *From (12) and the definition of we get
Then, using the total symmetry of and , we get the required result.

Corollary A.2. *The tensor field ** has the following symmetry properties* (the difference between these properties and those of Proposition 3.1 of [2] is explained by our different choice of the order of arguments in the Hessian curvature):
*
The same symmetries also hold for .*

*Proof. *Use the expressions (A.1) and (13).

*Remark A.3. *Usually, the expression of the Riemannian curvature contains the second order derivatives of the metric. But, formula (A.1) shows that the Riemannian curvature of a Hessian metric contains only the first order derivatives. This phenomenon was studied in [7].

A comparison with Riemannian geometry, again, suggests a notion of* Hessian sectional curvature* [2]. As a matter of fact, the latter is an invariant associated with a quadratic cone of the tangent space of a locally affine manifold endowed with a metric . In order to define it we need the following observation. Any 4-time covariant tensor field that has the symmetry properties (A.3) is equivalent to a quadratic form on , which is defined by
on the generators . Equivalently, with respect to the local basis , if is a symmetric, -contravariant tensor, .

*Definition A.4. *Let be a quadratic cone defined by , where and is a -covariant, symmetric, tensor with . Then, the* conical* (sectional [2])* Hessian curvature* of is
where is defined by (A.4).

The value of does not change under the multiplication of by a scalar. If is a generator of , the conical curvature of may be written (omitting ) as where has the same symmetry properties like .

Proposition A.5 (see [2]). *The conical curvature of a (pseudo)Hessian metric is independent of the cone; that is, , if and only if
**
Moreover, in this case, and, if , .*

*Proof. *If , the quadratic form vanishes and so does the corresponding symmetric bilinear form. This fact exactly is (A.8). Furthermore, if (A.8) holds, (14) implies that the Riemannian, sectional curvature is and Schur’s theorem (Theorem in [5]) shows that, if , .

Hessian metrics of constant conical curvature were studied in [2] and more recently in [14].

*Open Problem.* If the metric is positive definite, so is the corresponding form given by (A.4) and we can consider* principal cones * and* principal conical curvatures *, defined by the eigenvectors and the eigenvalues of with respect to . It would be interesting to study “ombilical” Hessian manifolds defined by the equality of all the principal conical curvatures .

#### Conflict of Interests

The author declares that there is no conflict of interests regarding the publication of this paper.

#### References

- S. Y. Cheng and S. T. Yau, “The real Monge-Ampère equation and affine flat structures,” in
*Proceedings of the Beijing Symposium on Differential Geometry and Differential Equations*, pp. 339–370, Science Press, Beijing, China, 1980. View at: Google Scholar | MathSciNet - H. Shima,
*The Geometry of Hessian Structures*, World Scientific, Singapore, 2007. View at: Publisher Site | MathSciNet - B. Totaro, “The curvature of a Hessian metric,”
*International Journal of Mathematics*, vol. 15, no. 4, pp. 369–391, 2004. View at: Publisher Site | Google Scholar | Zentralblatt MATH | MathSciNet - I. Bucataru and R. Miron,
*Finsler-Lagrange Geometry*, Editura Academiei Romane, Bucharest, Romania, 2007. View at: MathSciNet - S. Kobayashi and K. Nomizu,
*Foundations of Differential Geometry*, vol. 1, Wiley-Interscience, New York, NY, USA, 1963. View at: MathSciNet - I. Vaisman, “Lagrange geometry on tangent manifolds,”
*International Journal of Mathematics and Mathematical Sciences*, vol. 2003, no. 51, pp. 3241–3266, 2003. View at: Publisher Site | Google Scholar | Zentralblatt MATH | MathSciNet - J. J. Duistermaat, “On Hessian Riemannian structures,”
*The Asian Journal of Mathematics*, vol. 5, no. 1, pp. 79–91, 2001. View at: Google Scholar | Zentralblatt MATH | MathSciNet - A. Bejancu, “Schouten-Van Kampen and Vrănceanu connections on foliated manifolds,”
*Analele Ştiintifice ale Universităţii “Al. I.Cuza” din Iaşi, Matematică*, vol. 52, no. 1, pp. 37–60, 2006. View at: Google Scholar | MathSciNet - G. Vrănceanu, “Sur quelques points de la theorie des espaces non holonomes,”
*Buletinul Facultăţii de Ştiinţe din Cernăuţi*, vol. 5, pp. 177–205, 1931. View at: Google Scholar - I. Vaisman, “Variétés riemanniennes feuilletées,”
*Czechoslovak Mathematical Journal*, vol. 21, no. 96, pp. 46–75, 1971. View at: Google Scholar | Zentralblatt MATH | MathSciNet - I. Vaisman,
*Cohomology and Differential Forms*, Marcel Dekker, New York, NY, USA, 1973. View at: MathSciNet - D. Bao, S.-S. Chern, and Z. Shen,
*An Introduction to Riemann-Finsler Geometry*, vol. 200 of*Graduate Texts in Mathematics*, Springer, New York, NY, USA, 2000. View at: Publisher Site | MathSciNet - I. Vaisman, “Generalized CRF-structures,”
*Geometriae Dedicata*, vol. 133, no. 1, pp. 129–154, 2008. View at: Publisher Site | Google Scholar | Zentralblatt MATH | MathSciNet - H. Furuhata and T. Kurose, “Hessian manifolds of nonpositive constant Hessian sectional curvature,”
*The Tohoku Mathematical Journal*, vol. 65, no. 1, pp. 31–42, 2013. View at: Publisher Site | Google Scholar | Zentralblatt MATH | MathSciNet

#### Copyright

Copyright © 2014 Izu Vaisman. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.