A Local Classification of Some Special -Metrics of Constant Flag Curvature
We classify some special Finsler metrics of constant flag curvature on a manifold of dimension .
One of the important problems in Finsler geometry is to study and characterize Finsler metrics of constant flag curvature, which is the generalization of sectional curvature in Riemannian geometry. The local structure of Finsler metrics of constant flag curvature has been historically mysterious and their classification seems to be far from being solved.
The -metrics are an important class of Finsler metrics including Randers metrics as the simplest class. By making use of navigation problem, Bao et al. gave a local classification of Randers metrics with constant flag curvature . Recently, Zhou has classified that the -metrics with constant flag curvature in the following form, are locally projectively flat .
Lately, Shen and Zhao have studied projectively flat -metrics where is a constant and is a nonzero constant, and they proved such projectively flat Finsler metrics with constant flag curvature must be locally Minkowskian . Hence, one natural problem is to consider the classification of such metrics with constant flag curvature. In this paper, we prove the following rigidity result.
Theorem 1. Let , where the 1-form is nonzero, is a constant, and is a nonzero constant, be an -metric on a manifold of dimension . Suppose that is of constant flag curvature; then it must be locally Minkowskian.
Let be a Finsler metric on an -dimensional manifold and the geodesic coefficients of , which are defined by For any and , the Riemann curvature is defined by
-metrics were first introduced by Matsumoto . They are expressed in the following form: where is a Riemannian metric and is a 1-form. is a smooth positive function satisfying
For any flag , where , the flag curvature is defined by When is Riemannian, is independent of . It is just the sectional curvature of in Riemannian geometry. is said to be of scalar curvature if, for any , the flag curvature is independent of containing that is equivalent to the following system of equations in a local coordinate system in , If is a constant, then is said to be of constant flag curvature.
Let where “” denotes the covariant derivative with respect to the Levi-Civita connection of . Clearly is closed if and only if . Moreover, we denote where and . Let and be the geodesic coefficients of and , respectively. Then we have the following.
Lemma 2 (see ). For an -metric , , the geodesic coefficients are given by where Here and .
Let , ; one has the following.
Proposition 3 (see ). For any -metric , , the Riemannian curvature is given by where
Proposition 4 (see ). For an -metric , the Ricci curvature of is related to the Ricci curvature of by where
Remark 5. In Proposition 4, we have corrected some terms in the formulas for coefficients and , which are not printed.
Definition 6 (see ). Let where are the spray coefficients of . The tensor is called Douglas tensor. A Finsler metric is called Douglas metric if the Douglas tensor vanishes.
Note that an -metric is a Douglas metric if and only if holds for some scalar function .
Definition 7 (see ). Put where . Then is a tensor on . it is called the Weyl curvature tensor.
It is Hilbert’s fourth problem in the regular case to study and characterize Finsler metrics on an open domain whose geodesics are straight lines. Finsler metrics with this property are called projectively flat metrics. A famous theorem of Douglas is in the following.
Theorem 8 (see ). A Finsler metric on a manifold (dim ) is locally projectively flat if and only if and .
3. Proof of Theorem 1
In this section, we will prove Theorem 1. First, we will study the following lemma, because Lemma 4.1  is found to have some wrong. For example, when , does not satisfy the definition of -metrics at .
Lemma 9. is a Finsler metric if and only if one of the following holds.(a)If , then and .(b)If , then and , where , for any .
Direct computations yield From the definition of , we know that, for all , must satisfy That is, Particularly, when and , it is easy to see that
Now we first discuss the second inequality of (22). It is equivalent to the following two cases.
Let ; it is easy to see that and are quadratic functions with respect to .
Case 1. When (I).
By the graph of quadratic function and (23), we find that there is no such that and always hold for all . Consider (II) .
By the graph of quadratic function and (23), we conclude the same result with (I).
In any case, Case 1 cannot exist for all .
Case 2. When (III).
By the graph of quadratic function and (23), the above system of inequality always holds if and only if . Consider (IV) .
Similarly, the above system of inequality always holds if and only if .
In the following, we will discuss the condition satisfying the first inequality of (22). Since
let . Consider (1) .
In this case, by (III), we know that . By the graph and monotonicity of quadratic functions, we find Hence, for all , (26) always holds if and only if . Moreover, if , we have . If , holds forever for all . If , we get .
In a word, . Consider (2) .
In this case, by (IV), we know that . By the same way as above, we have
Hence, for all , (26) always holds if and only if . Furthermore,
if , we have . If , holds forever for all . If , we get .
In any case, . We complete the proof of the lemma.
Lemma 10. Let be a Finsler metric on a manifold , where is a constant and is a nonzero constant. If is of constant flag curvature, then it must satisfy(a),(b),(c),
where is a smooth function on .
Proof. By a direct computation, we have
Since is of constant flag curvature, is also of constant Ricci curvature; that is, the following system of equations holds
By Proposition 4, we can calculate by Maple where and are polynomials of .
Plugging (31) into (30) and multiplying it by yield It is easy to see that Hence, From (31) we find that By Lemma 9, we need to divide two cases. Consider (1) .
In this case, we know that , where . Hence, .
For we get Simplifying (38) by Maple yields Since , . Moreover, we note that . Otherwise, by Lemma 9, is not a Finsler metric.
From (39), and Zhou’s Lemma 4.1 , we obtain that where and are smooth functions on a manifold .
It is easy to see that The above equation holds if and only if Therefore, we have where is a smooth function on a manifold . Consider (2) .
In this case, we know that , where . Hence, . Moreover, is irreducible with respect to .
Simplifying (36) directly by Maple yields By Lemma 9, we note that . The above equation holds if and only if or where and are smooth functions on a manifold .
In any case, it is easy to find that . Hence, , where is a smooth function on a manifold .
Plugging (31) and into (30) and multiplying it by , we have where Simplifying (47) by Maple, we have So
Lemma 11. Let be a Finsler metric on an -dimensional manifold , where is a constant and is a nonzero constant. Suppose that is of constant flag curvature; then is closed and .
Proof. By Lemma 10, we have
where is a smooth function on an -dimensional manifold .
Thus, by some computations, we have where .
Case 1 (). Plugging the above equations into the expression of in Proposition 3 and simplifying it by Maple, we have where are polynomials of , , , and .
Because has constant flag curvature, it is equivalent to
Substitute (53) into (54) and multiplying it by , we find where .
By the same way as Lemma 10, we conclude or ; that is, is closed.
Case 2 . By the same approach as the above, we still conclude that is closed.
Now we will prove .
Let . If , , has constant flag curvature , then has constant Ricci curvature. That means that . Hence, is an Einstein metric with Ricci constant .
Note that is a polynomial in of degree . According to Theorem 1.1 (see ), we know that it is Ricci flat. So .