S S symmetry Article Generalized Semi-Symmetric Non-Metric Connections of Non-Integrable Distributions Tong Wu and Yong Wang * School of Mathematics and Statistics, Northeast Normal University, Changchun 130024, China; [email protected] * Correspondence: [email protected] Abstract: In this work, the cases of non-integrable distributions in a Riemannian manifold with the first generalized semi-symmetric non-metric connection and the second generalized semi-symmetric non-metric connection are discussed. We obtain the Gauss, Codazzi, and Ricci equations in both cases. Moreover, Chen’s inequalities are also obtained in both cases. Some new examples based on non-integrable distributions in a Riemannian manifold with generalized semi-symmetric non-metric connections are proposed. Keywords: non-integrable distributions; semi-symmetric non-metric connections; Chen’s inequali- ties; Einstein distributions; distributions with constant scalar curvature MSC: 53C40; 53C42 1. Introduction In [1], the notion of a semi-symmetric metric connection on a Riemannian manifold was introduced by H. A. Hayden. Some properties of a Riemannian manifold endowed with a semi-symmetric metric connection were studied by K. Yano [2]. Later, the properties Citation: Wu, T.; Wang, Y. of the curvature tensor of a semi-symmetric metric connection in a Sasakian manifold Generalized Semi-Symmetric were also investigated by T. Imai [3,4]. Z. Nakao [5] studied the Gauss curvature equation Non-Metric Connections of and the Codazzi–Mainardi equation with respect to a semi-symmetric metric connection Non-Integrable Distributions. Symmetry 2021, 13, 79. https:// on a Riemannian manifold and a submanifold. The idea of studying the tangent bundle doi.org/10.3390/sym13010079 of a hypersurface with semi-symmetric metric connections was presented by Gozutok and Esin [6]. In [7], Demirbag investigated the properties of a weakly Ricci-symmetric Received: 16 December 2020 manifold admitting a semi-symmetric metric connection. N. S. Agashe and M. R. Chafle Accepted: 2 January 2021 showed some properties of submanifolds of a Riemannian manifold with a semi-symmetric Published: 5 January 2021 non-metric connection in [8,9]. In [10,11], the study of non-integrable distributions, as a generalized version of distributions, was initiated by Synge. In [12], a regular distribution Publisher’s Note: MDPI stays neu- was shown in a Riemannian manifold. tral with regard to jurisdictional clai- Besides this, in [13–15], an important inequality was established by B. Y. Chen, called ms in published maps and institutio- Chen inequality. In geometry, by studying different submanifolds in various ambient nal affiliations. spaces, we can obtain similar results. In [16,17], Mihai and O¨ zgu¨ presented the relationships between the mean curvature associated with the semi-symmetric metric connection, scalar, and sectional curvatures and the k-Ricci curvature. In this paper, we obtain the Chen inequalities of non-integrable distributions of real-space forms endowed with the first Copyright: © 2021 by the authors. Li- censee MDPI, Basel, Switzerland. generalized semi-symmetric non-metric connection and the second generalized semi- This article is an open access article symmetric non-metric connection. distributed under the terms and con- In the literature, we find several works that were conducted with Einstein manifolds ditions of the Creative Commons At- and manifolds involving a constant scalar curvature. In [18], Dobarro and Unal studied tribution (CC BY) license (https:// Ricci-flat and Einstein Lorentzian multiply-warped products and constant scalar curvatures creativecommons.org/licenses/by/ for this class of warped products. In [19–21], the authors obtained some results with Einstein 4.0/). warped-product manifolds with a semi-symmetric non-metric connection. Symmetry 2021, 13, 79. https://doi.org/10.3390/sym13010079 https://www.mdpi.com/journal/symmetry Symmetry 2021, 13, 79 2 of 19 In Section2, we obtain the Gauss, Codazzi, and Ricci equations for non-integrable distributions with the first generalized semi-symmetric non-metric connection by estab- lishing the Gauss formula and the Weingarten formula. Meanwhile, the result of the Chen inequality is presented. In Section3, we obtain the Gauss, Codazzi, and Ricci equations for non-integrable distributions by establishing the Gauss formula and the Weingarten formula and the second generalized semi-symmetric non-metric connection. Meanwhile, we obtain the result of the Chen inequality. Finally, in Section4, some examples based on non-integrable distributions in a Riemannian manifold with generalized semi-symmetric non-metric connections are presented. 2. Non-Integrable Distributions with the First Generalized Semi-Symmetric Non-Metric Connection Let (M, g) be a m-dimensional smooth Riemannian manifold, where g is the Rieman- nian metric and r is the Levi–Civita connection on (M, g). For X, Y 2 G(M), denote rXY the covariant derivative of Y with respect to X and represent by G(M) the C¥(M)-module of vector fields on M. Definition 1. If there are X, Y 2 G(D) such that [X, Y] is not in G(D), we say that D is a non-integrable distribution, where D is a sub-bundle of the tangent bundle TM with a constant rank n and G(D) is the space of sections of D. ? Let gD be a metric tensor field in the distribution D and let gD be a metric tensor ? field in the orthogonal distribution to D, such that g = gD ⊕ gD . ? Definition 2. Let pD : TM ! D, pD : TM ! D? be the projections associated to the tangent D D D D D? D? bundle TM; then, rX Y = p (rXY) and [X, Y] = p ([X, Y]) and [X, Y] = p ([X, Y]) for any X, Y 2 G(D). By [12], we obtain D D D D r f XY = f rX Y, rX ( f Y) = X( f )Y + f rX Y, (1) where X, Y 2 G(D) and f 2 C¥(M). D D D D D? rX g = 0, T(X, Y) := rX Y − rY X − [X, Y] = −[X, Y] , (2) and D rXY = rX Y + B(X, Y). (3) D? where B(X, Y) = p rXY and B(X, Y) 6= B(Y, X). Definition 3. For any V 2 G(TM), let w be a 1-form satisfying w(V) = g(U, V), here U 2 ¥ G(TM) is a vector field. Let l1, l2 2 C (M), we give the definition of the first generalized semi-symmetric non-metric connection on M re XY = rXY + l1w(Y)X − l2g(X, Y)U. (4) ? ? ? Let UD = pDU and UD = pD U; then, U = UD + UD . Definition 4. Let D D D D? re XY = re X Y + Be(X, Y), re X Y = p re XY, Be(X, Y) = p re XY. (5) Symmetry 2021, 13, 79 3 of 19 Then, D D D D? re X Y = rX Y + l1w(Y)X − l2g(X, Y)U , Be(X, Y) = B(X, Y) − l2g(X, Y)U , (6) where Be(X, Y) is called the second fundamental form with the first generalized semi-symmetric non-metric connection. Then, by (2) and (6), we obtain D D re X (g )(Y, Z) = (l2 − l1)[g(X, Z)w(Y) + g(X, Y)w(Z)], (7) D D? Te (X, Y) = −[X, Y] + l1[w(Y)X − w(X)Y]. If D = TM, we obtain the following results: Theorem 1. If a linear connection re D : G(D) × G(D) ! G(D) on D satisfies Equation (7), then this connection is unique. 1 n We choose fE1, ... , Eng as an orthonormal basis of D and let He = n ∑i=1 Be(Ei, Ei) 2 ? 1 n G(D ) be the mean curvature vector associated to re on D. Similarly, let H = n ∑i=1 B(Ei, Ei); D? then, He = H − l2U . If He = 0, we say that D is minimal with the first generalized semi- symmetric non-metric connection re . If re g˙ g˙ = 0, we say that curve g is re -geodesic. If every re -geodesic with an initial condition in D is contained in D, we say that D is totally geodesic with the first generalized semi-symmetric non-metric connection re . 1 1 Let h(X, Y) = 2 [B(X, Y) + B(Y, X)] and eh(X, Y) = 2 [Be(X, Y) + Be(Y, X)]; then, ac- cording to [12], we obtain the following: Proposition 1. (1) If D is totally geodesic with respect to the first generalized semi-symmetric non-metric connection re , then Be(X, Y) is dissymmetrical. (2) When U 2 G(D),H = H,Ue 2 G(D), and vice versa. (3) If h = HgD (or eh = Hge D), then D is umbilical with respect to r (resp. re ). Proposition 2. If D is umbilical with respect to r, then D is umbilical with respect to re , and vice versa. D? D? Proof. For X, Y 2 D, by eh(X, Y) = h(X, Y) − l2g(X, Y)U and He = H − l2U , D D D? D then Hge (X, Y) = Hg (X, Y) − l2U g (X, Y). Therefore, we obtain Proposition 1. Thus, by Definition 4, we obtain re Xh = rXh + l1w(h)X, (8) where h 2 G(D?) and X 2 G(D). We define D D? g (Ah X, Y) := g (B(X, Y), h), (9) ? D? where Ah : G(D) ! G(D) is the shape operator with respect to r. Let LX h = p rXh; D ? then, rXh = p rXh + LX h, and so we can get the Weingarten formula with respect to r D ? p rXh = −Ah X, rXh = −Ah X + LX h, (10) ? ? ? ? where LX h : G(D) × G(D ) ! G(D ) is a metric connection on D along G(D). Let Aeh = (Ah − l1w(h))I; then, by (8) and (10), we have the Weingarten formula with respect to re ? re Xh = −Aeh X + LX h, (11) Symmetry 2021, 13, 79 4 of 19 Given X1, X2, X3 2 G(TM), we define the curvature tensor Re with respect to re R(X X )X = r r X − r r X − r X e 1, 2 3 : e X1 e X2 3 e X2 e X1 3 e [X1,X2] 3.