Optimal Approximations on Riemannian Manifolds

M.Toda and C.Udrişte

Abstract

The paper studies some new variational problems, extending certain applications of optimal approximations by piecewise smooth functions in computer vision.

For this aim, we introduce families of functionals as generalizations of those considered by D.Mumford and I.Shah [1], via the idea of p-energy [4]-[7]. Among these functionals, one is of the Finslerian type.

Mathematics Subject Classification: 49F20, 92A27, 53B40 Key words: Riemannian manifold, variational problems, (α, β)-Finsler metric

Sonar, radar or laser ”range” data, involve a function φ(x, y) of two variables that represents the distance from a fixed point P in direction (x, y) to the nearest solid object. Another example using a function φ(x, y) of two variables is the one of a lens at P focussing the light on a planar domain R of Cartesian coordinates x, y, where φ(x, y) is the intensity of the light signal striking R at the point (x, y). The function φ(x, y) defined on the planar domain R is called an image. In [1], authors study the problem of appropriately decomposing the planar domain R. In this scope, the authors introduced certain variational problems. Obviously, in this context, the image φ(x, y) presents discontinuities caused by the location of the bodies.

Taking into account that in practice R is not always a planar domain (it might be a lens, a sphere, a parabolic surface and so on) and on the other side that the image may be reconverted by a regular transformation, we consider R as a twodimensional Riemannian space endowed with the Riemannian metric g(x, y). The set R is a bounded and the function φ(x, y) is modeled by a set of smooth functions Ψ_{i} respectively defined on disjoint connected open subsets R_{i} of the domain R, each one with a piecewise smooth boundary.

Let Γ be the union of the parts of the boundaries of R_{i} inside R. The decomposition R = R_{1} ∪ R_{2} ∪ … ∪ R_{n} ∪ Γ must be such that φ varies smoothly and slowly within each R_{i} and varies discontinuously across most of the set Γ. We look for an optimal approximation of φ(x, y) by piecewise smooth functions Ψ(x, y) such that Ψ_{i} = Ψ|_{R_{i}} be differentiable. For this aim, we generalize the functional E from [1],

Balkan Journal of Geometry and Its Applications, Vol.4, No.1, 1999, pp. 135-144 \begin{matrix}c \\ ◯ \end{matrix}Balkan Society of Geometers, Geometry Balkan Press

E(Ψ, Γ) = μ ∫ ∫_{R}(Ψ − φ)^{2}dxdy + ∫ ∫_{R−Γ} ||grad Ψ||^{2}dxdy + ν|Γ|,

of a planar domain, by introducing the functional

(1)  E_{p}(Ψ, Γ) = μ ∫ ∫_{R} |Ψ − φ|^{2p}dσ + ∫ ∫_{R−Γ} ||grad Ψ||^{2p}dσ + νl_{p}(Γ),

where Ψ is assumed to be differentiable on ∪R_{i} (but can be discontinuous on Γ), p a real positive number, dσ = \sqrt{det g}dx^{1} ∧ dx^{2} is the element of area, μ, ν are positive real numbers,

l_{p}(γ) = ∫_{0}^{1} ||γ̇(t)||^{p}dt = ∫_{0}^{1} [\sqrt{g_{ij}(x^{1}, x^{2})ẋ^{i}ẋ^{j}}]^{p} dt

is the p-energy of any parametrized trajectory γ(t), and l_{p}(Γ) is the total p-energy of the arcs making up Γ. The case p = 1 was studied in [1].

The best approximation Ψ of φ is the one which minimizes E_{p}. Remarks. a) the first term of E_{p} requires that Ψ approximates φ.

b) the second term of E_{p} requires that Ψ (and hence φ) does not vary strongly in the sets R_{i}.

c) the third term of E_{p} requires that the curve Γ that accomplish these conditions be as short as possible.

First, we shall analyse the first variation for E_{p} with respect to Γ, considering just the case of a simple point P ∈ Γ, near which Γ moves. Such a point P lies on exactly one curve γ_{i} ⊂ Γ of class C^{1} which in a small neighbourhood U of P can be considered as a plane curve of type y = h(x), or x = h(y). Moreover, we consider the set U to be planar. We can deform γ_{i} to the curve

γ_{i}(t) : y = h(x) + tδh(x),

where δh is zero outside a small neighbourhood of P.

If t is small, the new curve γ_{i}(t) meets no curve γ′_{j}, j ≠ i except at its endpoints, and

Γ(t) = γ_{i}(t) ∪ ⋃_{j≠i} γ_{j}

is a deformation of Γ. We can no longer speak of leaving Ψ fixed while Γ moves, since Ψ is of class C^{1} on R − Γ but Ψ is usually discontinuous across Γ. Instead of this, we set

Ψ^{+} = Ψ|_{U^{+}}, U^{+} = {(x, y)|y > h(x)} ∩ U
Ψ^{−} = Ψ|_{U^{−}}, U^{−} = {(x, y)|y < h(x)} ∩ U.

We choose some C^{1} extensions of Ψ^{+} from U^{+} to U, and of Ψ^{−} from U^{−} to U. This is possible, since Ψ ∈ C^{1} on both sides of Γ at all the simple points of Γ.

Now let

Ψ^{t}(p) = \{ \begin{matrix}Ψ(P) & if & P ∉ U \\ extension of Ψ^{+} & if & P ∈ U, P above γ_{i}(t) \\ extension of Ψ^{−} & if & P ∈ U, P below γ_{i}(t). \end{matrix}

Optimal Approximations on Riemannian Manifolds 137 Consequently, E_{p}(Ψ^{t}, γ(t)) − E_{p}(Ψ, γ) = μ ∫ ∫_{U} {[(Ψ^{t} − φ)^{2}]^{p} − [(Ψ − φ)^{2}]} dxdy+ + ∫ ∫_{U−Γ(t)} ||grad Ψ^{t}||^{2p}dxdy − ∫ ∫_{U−Γ} ||grad Ψ||^{2p}dxdy+ +ν [l_{p}(γ_{i}(t)) − l_{p}(γ_{i})] = μ ∫_{I} (∫_{h(x)}^{h(x)+tδh(x)} {[(Ψ^{−} − φ)^{2}]^{p} − [(Ψ^{+} − φ)^{2}]^{p}} dy) dx+ + ∫_{I} (∫_{h(x)}^{h(x)+tδh(x)} [||grad Ψ^{−}||^{2p} − ||grad Ψ^{+}||^{2p}] dy) dx+ +ν ∫_{I} [(1 + ((h + tδh)′)^{2})^{\frac{p}{2}} − (1 + h^{′2})^{\frac{p}{2}} ] dx, where the symbol ′ indicates differentiation with respect to x. The calculation above implies

\frac{δE_{p}}{δγ} = lim_{t→0} \frac{E_{p}(Ψ^{t}, γ(t)) − E_{p}(Ψ, γ)}{t} =

= μ ∫_{I} {[(Ψ^{−} − φ)^{2}]^{p} − [(Ψ^{+} − φ)^{2}]^{p}}|_{y=h(x)} δhdx+

+ ∫_{I} [||grad Ψ^{−}||^{2p} − ||grad Ψ^{+}||^{2p}]|_{y=h(x)} δhdx + ν lim_{t→0} \frac{∫_{I} ∑_{n=1}^{∞} \frac{d^{n}F}{dt^{n}} (0) · \frac{t^{n}}{n!} dx}{t},

where

F (x, t) = (1 + ((h + tδh)′)^{2})^{\frac{p}{2}} .

Using integration by parts, the third term in the above sum, i.e.,

ν ∫_{I} 2h′(δh)′ \frac{p}{2}(1 + h^{′2})^{\frac{p}{2} −1}dx

becomes

−νp ∫_{I} h′′(1 + h^{′2})^{\frac{p}{2} −2}δh(1 + (p − 1)h^{′2})dx,

and then

\begin{aligned}\frac{δE_{p}}{δγ_{i}} & = & ∫_{γ_{i}}\{[μ|Ψ^{−} − φ|^{2p} + ||grad Ψ^{−}||^{2p}]|_{γ_{i}} − [μ|Ψ^{+} − φ|^{2p}+ \\ & + & ||grad Ψ^{+}||^{2p}]|_{γ_{i}} − νph′′(1 + (p − 1)h^{′2})(1 + h^{′2})^{\frac{p}{2} −2}\}\frac{δh}{\sqrt{1 + h^{′2}}} ds, \end{aligned}

where ds = \sqrt{1 + h^{′2}}dx is the arc element of γ_{i}. The coefficient \frac{δh}{\sqrt{1 + h^{′2}}} represents the displacement of the deformed Γ along the normal lines to the initial Γ.

At an extremum of E_{p} we have

(2)  \begin{aligned}[μ|Ψ^{+} − φ|^{2p} & + & ||grad Ψ^{+}||^{2p}] − [μ|Ψ^{−} − φ|^{2p} + ||grad Ψ^{−}||^{2p}]+ \\ & + & νph′′[(p − 1)h^{′2} + 1](1 + h^{′2})^{\frac{p}{2} −2} = 0. \end{aligned}

The terms in the first two square brackets can be interpreted as the p-energy density corresponding to the functional E_{p}, above and below the curbe, respectively. The term

curv_{p}(γ_{i}) = ph′′[(p − 1)h^{′2} + 1](1 + h^{′2})^{\frac{p}{2} −2}

shall be called the p-curvature of the curve γ_{i}. We notice that for p = 1, this term becomes

k = \frac{h′′(x)}{[1 + h^{′2}(x)]^{3/2}} ,

i.e., the usual curvature of the plane curve y = h(x), which exists and is bounded almost everywhere. Denoting

e_{p}(Ψ; x, y) = μ|Ψ − φ|^{2p} + ||grad Ψ(x, y)||^{2p},

(2) may be rewritten as

(e_{p}(Ψ^{+}) − e_{p}(Ψ^{−}) + ν · curv_{p}(γ_{i}))|_{γ_{i}} = 0.

Second, we study the first variation of E_{p} with respect to Ψ. In (1), let R be relative compact with piecewise smooth boundary, φ a continuous function on R̄, μ, ν positive constants, Ψ|_{R−Γ} ∈ C^{1} with continuous derivatives up to all boundary point and Γ made up of a finite set of C^{1} curves. Let us now fix Γ and φ and let vary Ψ in the conditions δΨ ∈ C^{1}, Ψ ∈ C^{2} on R − Γ. Then we find

E_{p}(Ψ + δΨ, Γ) − E_{p}(Ψ, Γ) = μ ∫ ∫_{R} {[(Ψ + tδΨ − φ)^{2}]^{p} − [(Ψ − φ)^{2}]^{p}} dσ+
+ ∫ ∫_{R−Γ}[||grad Ψ + t grad δΨ||^{2p} − ||grad Ψ||^{2p}]dσ =

= μ ∫ ∫_{R} ∑_{n=1}^{∞} \frac{d^{(n)}F}{dt^{n}} (0)\frac{t^{n}}{n!} dσ + ∫ ∫_{R−Γ} ∑_{n=1}^{∞} \frac{d^{(n)}G}{dt^{n}}(0)\frac{t^{n}}{n!} dσ,

where F (t) = |Ψ + tδΨ − φ|^{2p},

\begin{aligned}G(t) & = & ||grad Ψ + t grad δΨ||^{2p} = (||grad Ψ||^{2} + 2t < grad Ψ, grad δΨ > + \\ & + & t^{2}||grad δΨ||^{2})^{p}, \end{aligned}

\frac{δE_{p}}{δΨ} = lim_{t→0} \frac{E_{p}(Ψ + tδΨ, Γ) − E_{p}(Ψ, Γ)}{t} = 2pμ ∫ ∫_{R} δΨ[(Ψ − φ)^{2}]^{p−1}(Ψ − φ)dσ+
+ 2p ∫ ∫_{R−Γ} < grad Ψ, grad δψ > ||grad Ψ||^{2(p−1)}dσ =

= 2p {μ ∫ ∫_{R} δΨsgn(Ψ − φ)|Ψ − φ|^{2p−1}dσ+

+ ∫ ∫_{R−Γ} < ||grad Ψ||^{2(p−1)}grad Ψ, grad δΨ > dσ} .

We obtain

\frac{1}{2p} \frac{δE_{p}}{δΨ} = ∫ ∫_{R−Γ} δΨ[μsgn(Ψ − φ)|Ψ − φ|^{2p−1}−
− div(||grad Ψ||^{2(p−1)}grad Ψ)]dσ+
+ ∫_{∂(R−Γ)} < n⃗, δΨ||gradψ||^{2(p−1)}gradΨ > ds,

using the Gauss-Ostrogradski formula.

Further, denoting B = ∂(R − Γ), we obtain by direct calculation

\frac{1}{2p} \frac{δE_{p}}{δΨ} = ∫ ∫_{R−Γ} δΨ[μsgn(Ψ − φ)|Ψ − φ|^{2p−1}−
− div(||grad Ψ||^{2(p−1)}grad Ψ)]dσ+
+ ∫_{B} δΨ||gradψ||^{2(p−1)} \frac{∂Ψ}{∂n}ds.

In the following, we shall apply the fundamental lemma of Variational Calculus twice. Taking ∂Ψ to be a test function, which is non-zero near one point of R − Γ and zero elswere, we deduce that Ψ satisfies:

(3)  div (||grad Ψ||^{2(p−1)}grad Ψ) = μsgn(Ψ − φ)|Ψ − φ|^{2p−1} on R − Γ;

(4)  ||grad Ψ||^{2(p−1)} \frac{∂Ψ}{∂n} = 0,

on ∂R and on the two sides γ_{i}^{±} of each γ_{i}.

Obviously, grad Ψ may be zero only at the singular points of B. Excepting them,

we get \frac{∂Ψ}{∂n} = 0 on ∂R and on the two sides γ_{i}^{±} of each γ_{i} . Remark. For p = 1, the PDE (3) becomes the Poisson equation ΔΨ = μ(Ψ − φ) on R − Γ, and (4) becomes the boundary condition

(4′)  \frac{∂Ψ}{∂n} = 0 on ∂R ⋃ (⋃_{i} γ_{i}^{t}) .

So, for p = 1, we obtain the Neumann problem (3), (4′).

By analogy with [1], it can be shown that the natural limit functional of E_{p} as μ → 0 is

(5)  E_{p}^{0}(Γ) = ∑_{i} ∫ ∫_{R_{i}} [(φ − mean_{R_{i}}φ)^{2}]^{p} dσ + ν_{0}l_{p}(Γ),

because, when Γ is fixed and μ → 0, the function Ψ which minimizes E_{p} tends to the piecewise constant limit Ψ|_{R_{i}} = mean_{R_{i}}φ.

Using geometric measure theory, it can be shown that the problem of minimizing E_{p}^{0} is well-posed: for any continuous φ, there exists a set Γ made up of a finite number of singular points joined by a finite set of E_{p}^{0}-minimizing C^{2}-arcs.

It is shown that E_{p}^{0} may be viewed as a modification of the usual 2-dimensional Plateau problem functional, of fixed length Γ, by an external force term that keeps the minimal surfaces R_{i} (in the Plateau problem) from collapsing. It is known that the two-dimensional Plateau problem has only rather uninteresting extrema with Γ as union of line-segments. If we should generalize E_{p}^{0}(Γ) for dimensions greater than two, the minimizing problem becomes difficult.

We recall the oriented Plateau problem:

Given an (n-1)-dimensional oriented submanifold A of an m-dimensional Riemannian manifold, find an n-dimensional orientable submanifold C of least area with ∂C = A.

In addition to this problem, our problem of minimizing E_{p}^{0} has ”pressure” terms, which renders it more difficult.

Let D be the domain of the body in R^{3} endowed with the Riemannian metric g. Let dω = \sqrt{det g}dx ∧ dy ∧ dz. We divide D by a finite set of smooth surfaces meeting only at their boundaries and thus we obtain open ”cells” D_{i}, i = \overline{1, n} in which we assume that the density varies smoothly and slowly. Thus,

D = D_{1} ∪ D_{2} ∪ … ∪ D_{n} ∪ ∑ .

In this case we can introduce

E_{p}^{0} = ∑_{i} ∫ ∫ ∫_{D_{i}} [(φ − mean_{D_{i}}φ)^{2}]^{p} dω + ν_{0}A_{p}(∑),

where ∑ is the union of the parts of the boundaries of D_{i} inside D, whose p-area is

A_{p} = ∫ ∫ [\sqrt{EG − F^{2}}]^{p} dσ,

and E, F, G are the coefficients of the metric induced on ∑ by g. By nontrivial calculations, we can obtain also the natural limit functional E_{p}^{∞}, as μ → ∞.

We are interested here especially in the case p = 1, for which

(6)  E^{∞}(Γ) = lim_{μ→∞} E_{1} = ∫_{Γ} [ν_{∞} − (\frac{∂φ}{∂n})^{2}] ds.

In this case, we notice that the integrand is a generalized Finsler metric. Concerning the minimization of E^{∞} over all Γ, this is not a well-posed problem in most cases. If ||grad φ||^{2} ≤ ν_{∞} everywhere, then E^{∞} ≥ 0 and the simple choice Γ = ∅ minimizes E^{∞}. But if ||grad φ||^{2} > ν_{∞} on a non-empty open set U, then consider Γ made up of many pieces of level curves of φ within U. On such Γ′s, E^{∞} tends to −∞. Minimizing E^{∞} on a suitable restricted class of Γ can be a well-posed problem (see [1]).

Now we focus on the geometrical meaning of the functional (6). The integrand is the generalized Finsler metric

\begin{aligned}L(x^{1}, x^{2}, dx^{1}, dz^{2}) & = & [ν_{∞}− < n⃗, grad φ >^{2}] ds = \\ & = & \{ν_{∞} − [g_{ij}n^{i} (g^{jk} \frac{∂φ}{∂x^{k}} )]^{2}\} \sqrt{g_{ij}dx^{i}dx^{j}}, \end{aligned}

where n^{i} are the direction cosines of the normal to the curve, i.e., n^{i} = \frac{ν^{i}}{\sqrt{g_{kl}ν^{k}ν^{l}}},

where ν^{i} satisfy the condition

g_{ij}ν^{i} \frac{dx^{j}}{ds} = 0.

Hence,

ν^{1} = −(g_{12}dx^{1} + g_{22}dx^{2}), ν^{2} = +(g_{11}dx^{1} + g_{12}dx^{2}),

modulo a certain scaling. One can check that

g_{kl}ν^{k}ν^{l} = (det g) · g_{kl}dx^{k}dx^{l},

and consequently

\begin{aligned}L(x^{1}, x^{2}, dx^{1}, dx^{2}) & = & [ν_{∞} − (n^{i} \frac{∂φ}{∂x^{i}} )^{2}] \sqrt{g_{kl}dx^{k}dx^{l}} = \\ & = & [ν_{∞} − \frac{(ν^{i} \frac{∂φ}{∂x^{i}} )^{2}}{(det g)g_{kl}dx^{k}dx^{l}} ] \sqrt{g_{kl}dx^{k}dx^{l}}. \end{aligned}

Denoting y^{1} = dx^{1}, y^{2} = dx^{2}, we notice that L(x^{1}, x^{2}, dx^{1}, dx^{2}) can be rewritten as

(7)  L(α, β) = ν_{∞}α − \frac{β^{2}}{α} ,

where

α = α(x^{1}, x^{2}, y^{1}, y^{2}) = \sqrt{g_{kl}y^{k}y^{l}},

and

β(x, y) = b_{i}(x)y^{i},

with

b_{1} = \frac{1}{\sqrt{Δ}} (g_{11} \frac{∂φ}{∂x^{2}} − g_{12} \frac{∂φ}{∂x^{1}} ) ,

b_{2} = \frac{1}{\sqrt{Δ}} (g_{12} \frac{∂φ}{∂x^{2}} − g_{22} \frac{∂φ}{∂x^{1}} ) , Δ = det g.

We can see that

L(x^{1}, x^{2}, ty^{1}, ty^{2}) = |t|L(x^{1}, x^{2}, y^{1}, y^{2}).

Thus, E^{∞}(Γ) = ∫_{Γ} L(α, β). According to [2], we have the following Definition. Let α(x, y) = \sqrt{g_{ij}(x)y^{i}y^{j}} be a Riemannian metric and β(x, y) = b_{i}(x)y^{i} a 1-form. Any 1-homogeneous Finsler metric of the form L(α, β) is called an (α, β)- metric. Notice. If g is the Euclidean metric in the plane, then

n = \frac{(−dx^{2}, dx^{1})}{\sqrt{(dx^{1})^{2} + (dx^{2})^{2}}} , b_{1} = \frac{∂φ}{∂x^{2}} , b_{2} = −\frac{∂φ}{∂x^{1}}

L = \frac{ν_{∞}[(dx^{1})^{2} + (dx^{2})^{2}] − (\frac{∂φ}{∂x^{1}} dx^{2} − \frac{∂φ}{∂x^{2}} dx^{1})^{2}}{\sqrt{(dx^{1})^{2} + (dx^{2})^{2}}} .

Our metric (7) is among the (α, β) − metrics. Though it is neither a Randers

metric L_{R}(α, β) = α + β, nor a Kropina metric L_{K}(α, β) = \frac{α^{2}}{β} + β, we have

L = ν_{∞}α − L_{−1;2},

where

L_{p;q} = α^{p}β^{q}, p + q = 1

are extensively studied metrics.

In [2], Matsumoto presented a way of easier calculating the fundamental tensor g̃ of the Finsler space when the metric L is given. We apply this expression to our L. Starting from the definition (g̃ = ∇^{0}∇^{0}(L^{2}/2)) of g̃, we use the formula

g̃(u, v) = [L(∇^{0}, ∇^{0}L)](u, v) + (∇^{0}L)(u) · (∇^{0}L)(v),
g̃_{ij} = L( ˙∂_{i}, ˙∂_{j}L) + ˙∂_{i}L ˙∂_{j}L,

where ˙∂_{i} = \frac{∂}{∂y_{i}} and so we find

g̃_{ij} = pg_{ij} + q_{0}b_{i}b_{j} + q_{−1}(b_{i}Y_{j} + b_{j}Y_{i}) + q_{−2}Y_{i}Y_{j},

where we used

α = g_{ij}(x)y^{i}y^{j}, β = b_{i}(x)y^{i}, Y_{i} = g_{ij}y^{j}, h_{ij} = L( ˙∂_{i} ˙∂_{j}L)

p = \frac{L}{α}L_{α}, q_{0} = L L_{ββ}, q_{−1} = \frac{L}{α}L_{αβ}, q_{−2} = \frac{L}{α^{2}} (L_{αα} − \frac{L_{α}}{α} ) .

The subscripts of q_{0}, q_{−1}, q_{−2} are used to indicate the respective degrees of homogeneity.

By the homogeneity, we obtain two identities

p + q_{−1}β + q_{−2}α^{2} = 0, q_{0}β + q_{−1}α^{2} = 0.

We also denote

Y_{i} = g_{ij}y^{j} = L( ˙∂_{i}L) = pY_{i} + L L_{β}b_{i}.

From the above relations, the fundamental tensor is

g̃_{ij} = pg_{ij} + p_{0}b_{i}b_{j} + p_{−1}(b_{i}Y_{j} + b_{j}Y_{i}) + p_{−2}Y_{i}Y_{j},

where the coefficients are: p as above and

p_{0} = q_{0} + (L_{β})^{2} = L L_{ββ} + (L_{β})^{2},

p_{−1} = q_{−1} + \frac{p}{L}L_{β} = \frac{1}{α}(L L_{αβ} + L_{α}L_{β}),

p_{−2} = q_{−2} + (\frac{p}{L})^{2} = \frac{1}{α^{2}} \{L L_{αα} + (L_{α})^{2} − \frac{L}{α}L_{α}\} .

For

L(α, β) = ν_{∞}α − \frac{β^{2}}{α} ,

we get

p = (ν_{∞} − \frac{β^{2}}{α^{2}} ) (ν_{∞} + \frac{β^{2}}{α^{2}} ) = ν_{∞}^{2} − (\frac{β}{α})^{4} ,

p_{0} = (ν_{∞}α − \frac{β^{2}}{α} ) (−\frac{2}{α}) + \frac{4β^{2}}{α^{2}} = −2ν_{∞} + \frac{6β^{2}}{α^{2}} ,

p_{−1} = \frac{1}{α} [(ν_{∞}α − \frac{β^{2}}{α} ) \frac{2β}{α^{2}} + (ν_{∞} + \frac{β^{2}}{α^{2}} ) (−\frac{2β}{α} )] = −\frac{4β^{3}}{α^{4}} ,

p_{−2} = \frac{1}{α^{2}} [(ν_{∞}α − \frac{β^{2}}{α} ) (−\frac{2β^{2}}{α^{3}} ) + (ν_{∞} + \frac{β^{2}}{α^{2}} )^{2} − ν_{∞}^{2} + \frac{β^{4}}{α^{4}} ] = \frac{4β^{4}}{α^{6}} .

Hence,

g̃_{ij} = [ν_{∞}^{2} − (\frac{β}{α})^{4}] g_{ij} = (\frac{6β^{2}}{α^{2}} − 2ν_{∞}) b_{i}b_{j} − \frac{4β^{3}}{α^{4}} (b_{i}Y_{j} + b_{j}Y_{i}) + \frac{4β^{4}}{α^{6}} Y_{i}Y_{j}.

By direct calculation we can state when g̃_{ij} is positive definite.

We remind that in a Finsler space with an (α, β)-metric, we have a linear connection (Γ_{jk}^{i}(x)) (that is, the Riemannian connection of the Riemannian metric α) and therefore we obtain a Finsler connection (Γ_{jk}^{i}, N_{j}^{i} = y^{k}Γ_{jk}^{i}, C_{jk}^{i}), where C_{jk}^{i} are usual quantities [2]. Making use of this Finsler connection, Hashiguchi and Ichijyo developed in [3] an interesting theory and presented examples of Berwald and Wagner spaces among Finsler spaces with (α, β)-metric. The study of the Finsler space with the metric (4) is the subject of a forecoming paper.

References

[1] D.Mumford, J.Shah, Optimal Approximations by Piecewise Smooth Functions and Associated Variational Problems Communications on Pure and Applied Mathematics, 42, (1989), 577-685.

[2] M.Matsumoto, Foundations of Finsler Geometry and Special Finsler Spaces, Karseisha, Otsu, 1986.

[3] M.Hashiguchi, K.Ichijyo, On Some Special (α, β)-Metrics, Preliminary Report.

[4] C.Udrişte, Properties of a functional defined on the path space of a Riemann manifold, Analele Universităţii Bucureşti, (1978), 97-105.

[5] C.Udrişte, Properties of a family of functionals, Proceedings of the National Seminar on Finsler Spaces, Braşov (1980), 219-221.

[6] C.Udrişte, A finite dimensional approximation to Ω_{p}^{c}, Proceedings of the National Seminar on Finsler Spaces, Braşov (1983), 225-226.

[7] C.Udrişte, Convex functions and optimization methods on Riemannian manifolds, Mathematics and Its Applications, 297, Kluwer Academic Publishers, 1994.

University of Kansas Mathematics Department KUID 613495, M.Toda, G.T.A. 405 Snow Hall Lawrence KS 66045, USA

University Politehnica of Bucharest Department of Mathematics I Splaiul Independentei 313 77206 Bucharest, Romania email:udriste@mathem.pub.ro