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Bu makalenin asıl amacı alfa Kenmotsu psödo metrik manifoldlar üzerinde bazı eğrilik özelliklerini incelemektir. Özellikle bu tür manifoldlar üzerinde lokal simetri, global ϕ-simetri ve lokal ϕ-simetri gibi tensör koşulları bazı ek şartlar altında göz önüne alınmıştır. Ayrıca, η-Einstein ve Einstein manifoldlar için gerek ve yeter koşullar çalışılmıştır. Bundan başka, ξ-kesit ve ϕ-kesit eğrilikleri ile ilgili bazı sonuçlar alfa Kenmotsu psödo metrik manifoldlar üzerinde verilmiştir. Son olarak, makale alfa Kenmotsu psödo metrik manifoldlar için açıklayıcı bir örnekle sonlandırılmıştır., The aim of this paper is to investigate some curvature properties on alpha Kenmotsu pseudo metric manifolds. In particular, the tensor conditions such as locally symmetry, globally ϕ-symmetry and locally ϕ-symmetry under some additional conditions on such manifolds are considered. Also, the necessary and sufficient conditions for η-Einstein and Einstein manifolds are studied. Furthermore, some results are related to ξ-sectional and ϕ-sectional curvatures on alpha Kenmotsu pseudo metric manifolds are given. Finally, the paper is concluded with an illustrative example for alpha Kenmotsu pseudo metric manifolds.

An interesting class of submanifolds of almost Hermitian manifolds (\tilde M,\tilde g,J) is the class of slant submanifolds. Slant submanifolds were introduced by the first author in [6] as submanifolds M of (\tilde M,\tilde g,J) such that, for any nonzero vector X \in TpM, p \in M, the angle q(X) between JX and the tangent space TpM is independent of the choice of p\in M and X \in TpM. The first results on slant submanifolds were summarized in the book [7]. Since then slant submanifolds have been studied by many geometers. Many nice results on slant submanifolds have been obtained during the last two decades. The main purpose of this paper is to study pointwise slant submanifolds in almost Hermitian manifolds which extends slant submanifolds in a very natural way. Several basic results in this respect are proved in this paper.

We define new special curves in Euclidean 3-space which we call slant helices and conical geodesic curves. Those notions are generalizations of the notion of cylindrical helices. One of the results in this paper gives a classification of special developable surfaces under the condition of the existence of such a special curve as a geodesic. As a result, we consider geometric invariants of space curves. By using these invariants, we can estimate the order of contact with those special curves for general space curves. All arguments in this paper are straight forward and classical. However, there have been no papers which have investigated slant helices and conical geodesic curves so far as we know.

In this work, biharmonic hypersurfaces of index 2 in pseudo-Euclidean space E52 are studied under the assumption of having mean curvature H whose gradient ÑH is light-like, i.e. hÑH;ÑHi = 0 and ÑH 6= 0. In the first two sections, the problem is introduced and some basic definitions and formulas that we will use in other part of the paper are recalled. Moreover, all possible canonical forms of the shape operator of a hypersurface of index 2 are obtained. In the third section of this work, for each of these cases, some of geometrical properties of hypersurfaces is investigated. In particular, there are 2 possible canonical forms of the shape operator for a biharmonic hypersurface such that whose gradient ÑH is light-like are obtained. After that, the non-existance of biharmonic hypersurface of index 2 in pseudo-Euclidean space E52 with the light-like ÑH is proved. In the last section, the results from this work is summarized and the discussion part is given., Bu çalışmada, $\mathbb E^5_2$ yarı-Öklid uzayının indisi 2 olan biharmonik hiperyüzeyleri, $\nabla H$ gradyenti ışıksal olan $H$ ortalama eğriliğine sahip olmaları, yani $\left\langle \nabla H,\nabla H\right\rangle = 0$ ve $\nabla H\neq0$ koşullarının sağlanması varsayımı altında incelenmiştir. İlk iki bölümde problem tanıtılmış ve çalışmanın diğer bölümünde kullanılacak bazı temel tanım ve formüller hatırlatılmıştır. Ayrıca, $2$ indisli bir hiperyüzeyin şekil operatörlerinin tüm mümkün kanonik formları elde edilmiştir. Çalışmanın üçüncü bölümünde, bu durumların her biri için hiperyüzeylerin bazı geometrik özellikleri araştırılmıştır. Özellikle, $\nabla H$ gradyenti ışıksal olan biharmonik hiperyüzeyin şekil operatörünün $2$ olası kanonik formu olduğu elde edilmiştir. Hemen ardından, $\mathbb E^5_2$ yarı-Öklid uzayında indisi $2$ ve ortalama eğriliğinin gradyenti ışıksal olan bir biharmonik hiperyüzeyin olmadığı ispatlanmıştır. Son bölümdeise, çalışmadan elde edilen sonuçlar özetlenmiş ve tartışma bölümü verilmiştir.

Bu çalışmada, e^(ax+by) yoğunluklu E^3_1 Lorentz-Minkowski uzayında, ikisi aynı anda sıfır olmayan a ve b sabitlerinin durumlarına göre, ağırlıklı eğrilikleri sıfır olan spacelike ve timelike düzlemsel eğriler yardımıyla oluşturulan dönel yüzeyler ve regle yüzeyler çalışılmıştır., In the present paper, the surfaces of revolution and ruled surfaces which are constructed with the aid of spacelike and timelike planar curves with vanishing weighted curvatures in Lorentz-Minkowski space 𝐸13 with density 𝑒𝑎𝑥+𝑏𝑦 according to the cases of not all zero constants 𝑎 and 𝑏 are studied.

Helisler ve slant helisler, doğada ve bilimdesıklıkla karşılaştığımız, bilgisayar destekli geometrik tasarımdan fraktalgeometriye, kinematikten DNA çiftlerine, geniş bir kullanım alanına sahip olaneğrilerdir. Bu çalışmada, 3-boyutlu Öklid uzayında bir eğrinin asli normallergöstergesinin doğrultu eğrileri tanımlanmış ve bu eğrilerin Frenet vektörlerive eğrilikleri arasındaki ilişkiler bulunmuştur. Bu doğrultu eğrilerikullanılarak, küresel eğrilerden helis ve slant helisler oluşturmak içinkullanışlı yöntemler elde edilmiştir. Son olarak, konu ile ilgili bir örnek verilmiştir., Helices and slant helices are curves which areoften encountered in nature and science and have a wide range of uses includingcomputer aided geometric design, fractal geometry, kinematics, and DNA double. Inthis paper, direction curves of principal normal indicatrix of a curve inEuclidean 3-space are defined and some relationships between Frenet vectors andcurvatures of these curves are found. By using these direction curves, someuseful methods to construct helices and slant helices from spherical curves areobtained. Finally, a related example is given.

Bu tezin amacı, 4-boyutlu Walker manifoldlar üzerindeki hemen hemen kompleks yapılar ile Hermit metriklere çalışmaktır. Walker 4-manifoldlar üzerinde Hermit metrik olduğunu gösteren metotlar uygulanıp anlatılmıştır. Ayrıca, Walker 4-manifoldlar üzerinde hemen hemen Hermit ve hemen hemen Kähler yapılar için örnekler verilmiştir.Anahtar Kelimeler: Walker manifold, Kähler yapı, Hermit metrik, Kompleks yapı. The aim of the present paper is to study Hermitian metrics for given almost complex structure on 4–dimensional Walker manifolds. As an illustration, by applying the method show explicitly Hermitian metrics on Walker 4–manifolds. Also, give examples for almost Hermitian manifolds and almost Kähler structures on Walker 4–manifolds.Keywords: Walker manifold, Kähler structure, Hermitian metric, Complex structure. 44

Bu çalışmada, eğrisi eğrisinin bir involütü olarak alındığında involüt eğrisinin Frenet vektörleri, eğrilik ve torsiyonu eğrisinin Darboux ( Jean Gaston Darboux,1842-1917) vektörü ile binormal vektörü arasındaki açısına bağlı olarak verildi. Bu durumda involüt eğri boyunca oluşan -scroll’un Gauss eğriliği, ortalama eğriliği, I. ve II. temel formları yeniden hesaplanmıştır. Son olarak da helis eğrisi boyunca oluşan involut -scroll’ların mapple programı ile çizimi verilmiştir., In this paper, when  is considered as the involute of the  curve, Frenet vectors, curvature and torsion of  are given, respectively depending on the angle ,  which is between W Darboux vector and B binormal vector of  curve. In this case, Gaussian and mean curvatures, I. and II. Fundamental forms of B -scroll generated by involute curve have been calculated. Finally the involute B -scrolls generated by helix curve have drawn application.

Bu çalışmada, spacelike eğrisi time eğrisinin bir involütü olarak alındığında spacelikeinvolüt eğrisinin Frenet vektörleri, eğrilik ve torsiyonu timelike eğrisinin Darboux vektörü ile binormal vektörü arasındaki lorentzian açısına bağlı olarak verildi. Bu durumda involüt eğri boyunca oluşan B-scroll’un şekil operatörüne karşılık gelen matris, Gauss eğriliği, ortalama eğriliği, I. ve II. temel formları yeniden hesaplandı, In this paper, when  is considered as the spacelike involute of the  timelike curve, Frenet vectors, curvature and torsion of  are given, respectively depending on the angle ,  which is betwen W Darboux vector and B binormal vector of  curve. In this case, matrix corresponding to the transformation weingarten, Gausian and mean curvatures, I. and I. fundamental forms of B – scroll generated by involute curve have been calculated.

The main purpose of this paper is to show how to obtain rigidity theorems with the help of curvature invariants in submanifolds of a semi-Riemannian manifold. For this purpose, the bounded sectional curvature is introduced and some special submanifolds of r-lightlike submanifolds of a semi-Riemannian manifold are investigated.

In the present paper, we study hemi-slant submanifolds of a locally product Riemannian manifold. We prove that the anti-invariant distribution involved in the definition of hemi-slant submanifold is integrable and give some applications of this result. We get a necessary and sufficient condition for a proper hemi-slant submanifold to be a hemi-slant product. We also study these types of submanifolds with parallel canonical structures. Moreover, we give two characterization theorems for the totally umbilical proper hemi-slant submanifolds. Finally, we obtain a basic inequality involving Ricci curvature and the squared mean curvature of a hemi-slant submanifold of a certain type of locally product Riemannian manifolds.

In this paper, firstly, we define a W -direction curve and W -rectifying curve of a Frenet curve in 3-dimensional Euclidean space E3 by using the unit Darboux vector field W of the Frenet curve and give some characterizations together with the relationships between the curvatures of each associated curve. We also introduce a V -direction curve, which is associated with a curve lying on an oriented surface in E3. Later, some new associated curves of a Frenet curve are defined in E4.

In this paper, our aim is to study some properties of the tangent bundle with a deformed complete lift metric.

This paper is concerned with almost para-complex structures on Walker 4-manifolds. For these structures, we study some problems of Kahler manifolds. We also give an example of a flat almost para-complex manifold, which consists of a nonintegrable almost para-complex structure on Walker 4-manifolds.

In this paper, it has been showed that every single indicatrix of tangents and

The object of the present paper is to study generalized Tanaka-Webster connection on a Kenmotsu manifold. Some conditions for phi-conformally flat, phi-conharmonically flat, phi-concircularly flat, phi-projectively flat, phi-W2 flat and phi-pseudo projectively flat Kenmotsu manifolds with respect to generalized Tanaka-Webster connection are obtained., Bu çalışmada bir Kenmotsu manifold üzerinde genelleştirilmiş Tanaka-Webster konneksiyonu çalışıldı. Genelleştirilmiş Tanaka-Webster konneksiyonuna sahip phi-conformally flat, phi-conharmonically flat, phi-concircularly flat, phi-projectively flat, phi-W2 flat ve phi-pseudo projectively flat Kenmotsu manifoldlar için bazı şartlar elde edildi.

In this paper we study the non-geodesic non-null biharmonic curves in 3-dimensional hyperbolic Heisenberg group with a semi-Riemannian metric of index 2. We prove that all of the non-geodesic non-null biharmonic curves in such a 3-dimensional hyperbolic Heisenberg group are helices. Moreover, we obtain explicit parametric equations for non-geodesic non-null biharmonic curves and non-geodesic spacelike horizontal biharmonic curves, respectively. We also show that there do not exist non-geodesic timelike horizontal biharmonic curves in 3-dimensional hyperbolic Heisenberg group with a semi-Riemannian metric of index 2., Bu çalışmada indeksi 2 olan bir semi-Riemann metriğe sahip 3-boyutlu Heisenberg grubun jeodezik olmayan non-null biharmonik eğrileri çalışıldı. Bu şekildeki bir 3-boyutlu Heisenberg grubun jeodezik olmayan non-null biharmonik eğrilerinin helis olduğu ispatlandı. Ayrıca sırasıyla jeodezik olmayan non-null biharmonik eğriler ve jeodezik olmayan spacelike yatay biharmonik eğriler için açık parametrik denklemler elde edildi. İndeksi 2 olan bir semi-Riemann metriğe sahip 3-boyutlu Heisenberg grup üzerinde jeodezik olmayan timelike yatay biharmonik eğrilerin var olmadığı gösterildi.

In this paper, we obtain vertical, complete and horizontal lifts of paracomplex geometric structures on paracomplex manifolds to its tangent bundle. Also, we obtain integrability on paracomplex tangent bundle. © TÜBİTAK.

The Golden Ratio is a fascinating topic that continually generates new ideas. The main purpose of the present paper is to point out and find some applications of the Golden Ratio and of Fibonacci numbers in Differential Geometry. We study a structure defined on a class of Riemannian manifolds, called by us a Golden Structure. A Riemannian manifold endowed with a Golden Structure will be called a Golden Riemannian manifold. Precisely, we say that an (1,1)-tensor field \widetilde{P} on a m-dimensional Riemannian manifold (\widetilde{M}, \widetilde{g}) is a Golden Structure if it satisfies the equation \widetilde{P}2=\widetilde{P}+I (which is similar to that satisfied by the Golden Ratio f) where I stands for the (1,1) identity tensor field. First, we establish several properties of the Golden Structure. Then we show that a Golden Structure induces on every invariant submanifold a Golden Structure, too. This fact is illustrated on a product of spheres in an Euclidean space.

In this paper a generalization of the notion of regular curve is introduced. For such curves the concept of tangent line is investigated.

In this paper, we introduce the notion of a slant lightlike submanifold of an indefinite Kenmotsu manifold. We provide a non-trivial example and obtain necessary and sufficient conditions for the existence of a slant lightlike submanifold. Also, we give an example of a minimal slant lightlike submanifold of R29 and prove some characterization theorems.

In this paper we have obtained a general existence as well as uniqueness theorem for slant immersions into a Kenmotsu-space form.

In this paper, we discuss some geometric properties of almost contact metric submersions involving symplectic manifolds. We show that the structures of quasi-K-cosymplectic and quasi-Kenmotsu manifolds are related to (1, 2)-symplectic structures. For horizontally submersions of contact CR-submanifolds of quasi-K-cosymplectic and quasi-Kenmotsu manifolds, we study the principal characteristics and prove that their total spaces are CR-product. Curvature properties between curvatures of quasi-K-cosymplectic and quasi-Kenmotsu manifolds and the base spaces of such submersions are also established. We finally prove that, under a certain condition, the contact CR-submanifold of a quasi Kenmotsu manifold is locally a product of a totally geodesic leaf of an integrable horizontal distribution and a curve tangent to the normal distribution.

The homeomorphism problem is, given two compact n-manifolds, is there an algorithm to decide if the manifolds are homeomorphic or not. The homeomorphism problem has been solved for many important classes of 3-manifolds - especially those with embedded 2-sided incompressible surfaces (cf [12], [15], [16]), which are called Haken manifolds. It is also well-known that the homeomorphism problem is easily solvable for two 3-manifolds which admit geometries in the sense of Thurston [36], [31]. Hence the recognition problem, to decide if a 3-manifold has a geometric structure, is a significant problem. The recognition problem has been solved for all geometric classes, except for the class of small Seifert fibered spaces, which either have finite fundamental group or have fundamental groups which are extensions of Z by a triangle group and have finite abelianisation. Our aim in this paper is to give an algorithm to recognise these last classes of 3-manifolds, i.e to decide if a given 3-manifold is homeomorphic to one in this class. A completely different solution has been announced recently by Tao Li [22]. Also Perelman's announcement of a solution of the geometrisation conjecture would enable a complete solution of the homeomorphism problem; by identifying which geometric structure a given manifold admits. However it is worth noting that practical algorithms for the homeomorphism and recogntion problems, which can be implemented via software, are very useful for experimentation in 3-manifold topology. (See for example [5], [39]).

Let M be a smooth manifold with Finsler metric F, and let TM° be the slit tangent bundle of M with a generalized Riemannian metric G, which is induced by F. In this paper, we extract many natural foliations of (TM°,G) and study some of their geometric properties. Next we use this approach to obtain new characterizations of Finsler manifolds with positive constant curvature.

É. Cartan [2] has originally introduced a Cartan space, which is considered as dual of Finsler space. H. Rund [10], F. Brickell [1] and others studied the relation between these two spaces. The theory of Hamilton spaces was introduced and studied by R. Miron ([8|, [9]). He proved that Cartan space is a particular case of Hamilton space. T. Igrashi ([5], [6]) introduced the notion of the (a, b)-metric in Cartan spaces and obtained the metric tensor and the invariants r and r which characterize the special classes of Cartan spaces with (a, b)-metric. This paper presents a study of Cartan spaces with (a, b)-metric admitting h-metrical d-connection. We prove the conditions for these spaces to be locally Minkowski and conformally flat.

In this paper, the geometry of F-invariant submanifolds of a Kaehlerian product manifold is studied. The fundamental properties of these submanifolds are investigated such as pseudo umbilical, curvature invariant, totally geodesic, mixed geodesic submanifold and locally decomposable Riemannian product manifold.

In this paper, we investigate mixed QR-submanifolds in a quaternionic space form and pseudo umbilical QR-submanifold of a quaternionic space form under some additional condition. Finally we give a necessary condition for QR-submanifold of a quaternion Kaehler manifold such that \dim u \perp =1 to be a 3-quasi Sasakian Manifold.

The object of the present paper is to study the geometry of trans-Sasakian manifold when it is projectively semi-symmetric, Weyl semi-symmetric and concircularly semi-symmetric.

The aim of this paper is to construct a specific codimension-1 foliation of \( D2 \times S1 \) with one Reeb component, and to show that this foliation locally minimizes the volume among foliations with the boundary torus as a leaf.

The main purpose of the paper is to investigate geodesics on the tangent bundle with respect to the Cheeger-Gromoll metric.

In this paper, we study quadric surfaces in a Euclidean 3-space. Furthermore, we classify quadric surfaces in a Euclidean 3-space in terms of the Gaussian curvature and the mean curvature.

The object of this paper is to study f-recurrent Kenmotsu manifolds. Also three-dimensional locally f-recurrent Kenmotsu manifolds have been considered. Among others it is proved that a locally f-recurrent Kenmotsu spacetime is the Robertson-Walker spacetime. Finally we give a concrete example of a three-dimensional Kenmotsu manifold.

In the present paper we obtain sharp estimates for the squared norm of the second fundamental form in terms of the mapping function for contact 3-structure CR-warped products isometrically immersed in Sasakian space form.

In this paper, we generalize the geometry of the invariant submanifolds of Riemannian product manifold to the geometry of the invariant submanifolds of Riemannian warped product manifold. We investigate some properties of an invariant submanifolds of a Riemannian warped product manifold. We show that every invariant submanifold of the Riemannian warped product manifold is a Riemannian warped product manifold. Also, we give a theorem on the pseudo-umbilical invariant submanifold. Further, we obtain that integral manifolds on an invariant submanifold are curvature-invariant submanifolds. Finally, we give a necessary condititon on a totally umbilical invariant submanifold to be totally geodesic.

This paper is devoted to a study of timelike ruled surfaces in three dimensional Minkowski space obtained by a spacelike straight line moving along a timelike curve. The central point, the curve of striction and the distribution parameter of a timelike ruled surface in Minkowski 3-space are considered, and some theorems relating to their structure are obtained. In addition, some results about developable timelike ruled surfaces are also given.

In this paper we introduce the notions of semi-slant and bi-slant submanifolds of an almost contact 3-structure manifold. We give some examples and characterization theorems about these submanifolds. Moreover, the distributions of semi-slant submanifolds of 3-cosymplectic and 3-Sasakian manifolds are studied. ϕi)i=1;2;3 and the vector elds should be slant or invariant with respect to all of the ϕi's. Therefore, it is a generalization of invariant, anti-invariant, slant, semi-slant, and bi-slant submanifolds in almost contact metric 3-structures and we denote them by 3- semi-slant and 3-bi-slant submanifolds. Following the approaches of (3, 13), we characterized 3-bi-slant and 3-semi-slant submanifolds and studied geometric properties of distributions of these submanifolds where the ambient manifolds are 3-Sasakian or 3-cosymplectic. It should be noted that, in the denition of semi-slant

In this paper, the pitch and the angle of pitch of a closed nonnull ruled hypersurface whose generators are spacelike are calculated in Rk+21 .

Let Nn(4c) be the complex space form of constant holomorphic sectional curvature 4c, j: M \to Nn(4c) be an immersion of an n-dimensional Lagrangian manifold M in Nn(4c). Denote by S and H the square of the length of the second fundamental form and the mean curvature of M. Let r be the non-negative function on M defined by r2=S-nH2, Q be the function which assigns to each point of M the infimum of the Ricci curvature at the point. In this paper, we consider the variational problem for non-negative functional U(j)=\intMr2dv=\intM(S-nH2)dv. We call the critical points of U(j) the Extremal submanifold in complex space form Nn(4c). We shall get the new Euler-Lagrange equation of U(j) and prove some integral inequalities of Simons' type for n-dimensional compact Extremal Lagrangian submanifolds j: M \to Nn(4c) in the complex space form Nn(4c) in terms of r2, Q, H and give some rigidity and characterization Theorems.

In this paper, the induced Ricci tensor and the extrinsic scalar curvature on lightlike submanifolds are obtained. This scalar quantity extend the result given by C. Atindogbe in [1]. An example of extrinsic scalar curvature on one class of 2-degenerate manifolds is provided. We investigate lightlike submanifolds which are locally symmetric, semi-symmetric, Ricci semi-symmetric in indefinite spaces form. In the coisotropic case, we show that, under some conditions, these lightlike submanifolds are totally geodesic.

The aim in the present paper is to study some basic geometric properties of semi-slant submanifolds of a nearly Kaehler manifold.

In this paper, we investigate lightlike hypersurfaces which are semi-symmetric, Ricci semi-symmetric, parallel or semi-parallel in a semi-Euclidean space. We obtain that every screen conformal lightlike hypersurface of the Minkowski spacetime is semi-symmetric. For higher dimensions, we show that the semi-symmetry condition of a screen conformal lightlike hypersurface reduces to the semi-symmetry condition of a leaf of its screen distribution. We also obtain that semi-symmetric and Ricci semi-symmetric lightlike hypersurfaces are totally geodesic under certain conditions. Moreover, we show that there exist no non-totally geodesic parallel hypersurfaces in a Lorentzian space.

In this paper, we study coisotropic submanifolds of a semi-Riemannian manifold. We investigate the integrability condition of the screen distribution and give a necessary and sufficient condition on Ricci tensor of a coisotropic submanifold to be symmetric. Finally, we present some new theorems and results about totally umbilical coisotropic submanifolds of a semi-Riemannian manifold

The main result of the present paper establishes a stability property of the first eigenvalue of the associated problem which deals with the p(x)-Laplacian on Riemannian manifolds with Dirichlet boundary condition.

In this paper,QR-submanifolds of quaternion Kaehlerian manifolds with \dim n \perp =1 has been considered. It is shown that each QR-submanifold of quaternion Kaehlerian manifold with \dim n \perp =1 is a manifold with an almost contact 3-structure. We apply geometric theory of almost contact 3-structure to the classification of QR-submanifolds (resp.Real hypersurfaces) of quaternion Kaehler manifolds (resp.IR4m, m>1). Some results on integrability of an invariant distribution of a QR-submanifold and on the immersions of its leaves are also obtained.

The problem studied in this paper is related to the biharmonicity of a vector field from a Riemannian manifold (M,g) to its tangent bundle TM equipped with the Sasaki metric gs. We show that a vector field on a compact manifold is biharmonic if and only if is harmonic. We also investigate the biharmonicity of vector field of M, as a map from (M,g) to (TM,gs).

In this paper some B.-Y. Chen inequalities for slant submanifolds in quaternionic space forms are established.

In this paper, lightlike hypersurfaces of indefinite Kenmotsu space form are studied. Some characterizations of non-existence of lightlike hypersurfaces of indefinite Kenmotsu space form are given.

In this paper, we consider a class of Finsler metrics called generalized Berwald metrics which contains the class of Berwald metrics as a special case. We prove that every generalized Berwald metrics with non-zero scalar flag curvature or isotropic Berwald curvature is a Randers metric. Then we prove that on generalized Berwald metrics, the notions of generalized Landsberg and Landsberg curvatures are equivalent.

In this paper, we consider Finsler metrics defined by a Riemannian metric and a 1-form on a manifold. We study these metrics with vanishing S-curvature. We find some conditions under which such a Finsler metric is Berwaldian or locally Minkowskian.