We may use geodesic-arc angle to construct the geodesic-arc sector S{x} (l1 , l2 ) ⊂ TxM , where l1 , l2 ∈ TxM are two unit vectors issued from the origin 0 ∈ TxM (and pointed to the indicatrix).
Finsleroid gives rise to the angle-preserving connection
RN and h2(x) is the value of curvature of the indicatrix supported by point x (see (2.6)).
Finsleroid gives rise to the angle-preserving connection
At each point x ∈ M , the ratio AREAFinsleroid Indicatrix /V OLU M EFinsleroid proves to be of the universal value in each dimension N (is independent of h), so that the ratio is exactly the same as it holds in the Riemannian limit (that is, when h = 1).
Finsleroid gives rise to the angle-preserving connection
However, there exists a simple possibility to elucidate the global structure of the F F P D -Finsleroid indicatrix.
Finsleroid gives rise to the angle-preserving connection
The equality (3.7) manifests that the transformation (6.26) maps -Finsleroid indicatrix in regions of the sphere S {ζ } regions of the F F P D {x} .
Finsleroid gives rise to the angle-preserving connection
S {−} {x} may be used to yield two covering charts for the indicatrix; they can be characterized by the angle ranges indicated in (6.22).
Finsleroid gives rise to the angle-preserving connection
Similarly to the Riemannian case proper, we need two charts to cover the indicatrix.
Finsleroid gives rise to the angle-preserving connection
Under the conditions formulated in the preceding theorem, the indicatrix supported by a point x ∈ M is of the constant curvature h2 (x), such that RFinsleroid Indicatrix = C (x) with C (x) = h2 (x).
Finsleroid gives rise to the angle-preserving connection
Therefore, since we adhere at measuring the angle α by the geodesic arc-length on the indicatrix, from the above formulas (2.1)-(2.6) we are entitled to conclude the following.
Finsleroid gives rise to the angle-preserving connection
To surely recognize the validity of this theorem, it is suﬃcient to take a glance on the equality (3.9) which represents inﬁnitesimally the squared length of the geodesic arc on the indicatrix.
Finsleroid gives rise to the angle-preserving connection
If the indicatrix supported by a point x ∈ M is a space of constant curvature, then the conformal property (2.1) holds at the point x.
Finsleroid gives rise to the angle-preserving connection
The ﬁnal condition is (Z5): The indicatrix is closed and regular.
Finsleroid gives rise to the angle-preserving connection
Let the Finsleroid indicatrix IF {x} supported by a ﬁxed point x ∈ M be parameterized by means of a convenient variable set ua (for instance, we can take ua = ζ a/ζ N in regions with ζ N 6= 0, or ua = y a/yN , whenever yN 6= 0.
Finsleroid gives rise to the angle-preserving connection
We have taken into account the fact that the conformal factor κ , having been proposed by (2.5), equals 1/h(x) on the indicatrix.
Finsleroid gives rise to the angle-preserving connection
From this standpoint, it is easy to make the search into the curvature of the indicatrix.
Finsleroid gives rise to the angle-preserving connection
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