Quarter-Symmetric Metric Connection

• Quarter-symmetric metric connections are linear connections that preserve the metric while incorporating torsion defined by a 1-form and a (1,1)-tensor field.
• They generalize semi-symmetric connections and are crucial in studying geometric structures such as Hermitian, Sasakian, and submanifold configurations.
• Applications include curvature deformation analysis, classification of Einstein warped products, and derivation of sharp curvature inequalities in various manifolds.

A quarter-symmetric metric connection is a natural class of linear connections on differentiable manifolds whose torsion tensor is determined by a 1-form and a (1,1)-tensor field, and which preserve the underlying (pseudo-)Riemannian metric. These connections generalize the well-studied semi-symmetric connections by relaxing the symmetry required in the structure of the torsion, allowing for a wider class of geometric deformations. Quarter-symmetric metric connections play a central role in differential geometry, particularly in the study of Hermitian, almost Hermitian, Sasakian, and related geometric structures, as well as in modern submanifold theory and the classification of metric connections with prescribed torsion.

1. Definition and Canonical Formulation

Let $(M,g)$ be a (pseudo-)Riemannian manifold. A linear connection $\widetilde{\nabla}$ on $M$ is called a quarter-symmetric metric connection if it is metric, meaning $\widetilde{\nabla}g=0$, and its torsion $T$ has the form

$T(X,Y) = \eta(Y)QX - \eta(X)QY,$

where $\eta$ is a globally defined 1-form, $Q$ is a (1,1)-tensor field, and $X,Y$ are arbitrary vector fields on $M$. The metric preservation uniquely determines the connection: $\widetilde{\nabla}$0 with $\widetilde{\nabla}$1 the Levi-Civita connection and $\widetilde{\nabla}$2 the $\widetilde{\nabla}$3-dual of $\widetilde{\nabla}$4, i.e., $\widetilde{\nabla}$5 (Pal et al., 2022). This construction encompasses a range of geometric settings, including almost Hermitian, Kähler, (para-)Sasakian, and Lorentzian concircular manifolds.

A special and widely studied case is when $\widetilde{\nabla}$6; then $\widetilde{\nabla}$7, yielding the vectorial class of quarter-symmetric metric connections (Moroianu et al., 2020).

2. Structural Properties and Preservation Conditions

The connection $\widetilde{\nabla}$8 is compatible with the metric by construction. In geometric structures with additional tensors $\widetilde{\nabla}$9, e.g., the almost complex structure in an almost Hermitian manifold, one may ask for further preservation conditions. For instance, for $M$0 with $M$1, on an almost Hermitian manifold,

$M$2

and $M$3 always holds. However, $M$4 if and only if $M$5 is Kähler, i.e., $M$6 (Zlatanović et al., 2022). Thus, the quarter-symmetric metric connection preserving both $M$7 and the almost complex structure exists if and only if the manifold is Kähler.

Quarter-symmetric metric connections are characterized by their torsion being purely "quarter-symmetric," i.e., expressible entirely in terms of $M$8 and $M$9 with no symmetric or totally skew-symmetric part. This is in contrast to the general $\widetilde{\nabla}g=0$0-irreducible decomposition of torsion into vectorial, twistorial, and totally skew-symmetric components (Moroianu et al., 2020).

3. Curvature Tensors and Canonical Invariants

The curvature operator of a quarter-symmetric metric connection is a deformation of the Riemannian curvature, incorporating terms involving the 1-form $\widetilde{\nabla}g=0$1 and the tensor $\widetilde{\nabla}g=0$2. In general,

$\widetilde{\nabla}g=0$3

where $\widetilde{\nabla}g=0$4 is the Riemannian curvature, and $\widetilde{\nabla}g=0$5 are tensors constructed from $\widetilde{\nabla}g=0$6, $\widetilde{\nabla}g=0$7, and their derivatives (Pal et al., 2022).

In Kähler or almost Hermitian contexts, the presence of $\widetilde{\nabla}g=0$8 allows the construction of multiple curvature-like tensors. Specifically, six linearly independent curvature operators $\widetilde{\nabla}g=0$9 ($T$0), varying by the placement and contraction of derivatives of $T$1 and $T$2, can be written explicitly (Zlatanović et al., 2022). On a Kähler manifold, canonical invariants $T$3 can be constructed that are independent of the choice of generating 1-form $T$4. These invariants include:

• $T$5, coinciding with the Weyl projective curvature tensor $T$6 of $T$7,
• $T$8, related via explicit linear identities to $T$9 and the holomorphically projective tensor $T(X,Y) = \eta(Y)QX - \eta(X)QY,$0.

Table: Notable Invariants from Curvature Contractions on Kähler Manifolds (Zlatanović et al., 2022)

Invariant	Expression (schematic)	Geometric Role
$T(X,Y) = \eta(Y)QX - \eta(X)QY,$1	$T(X,Y) = \eta(Y)QX - \eta(X)QY,$2 + Ricci traces	Weyl projective curvature
$T(X,Y) = \eta(Y)QX - \eta(X)QY,$3	Combinations of $T(X,Y) = \eta(Y)QX - \eta(X)QY,$4	Relations with $T(X,Y) = \eta(Y)QX - \eta(X)QY,$5
$T(X,Y) = \eta(Y)QX - \eta(X)QY,$6	Combinations + traces	A-symmetries, hybrid tensors

These invariants play key roles in relating the geometry of quarter-symmetric metric connections to the classical Riemannian invariants of the base metric.

4. Special Instances: Sasakian, Para-Sasakian, and $T(X,Y) = \eta(Y)QX - \eta(X)QY,$7-Sasakian Manifolds

Quarter-symmetric metric connections naturally appear in almost contact geometry. For an $T(X,Y) = \eta(Y)QX - \eta(X)QY,$8-cosymplectic or (para-)Sasakian manifold $T(X,Y) = \eta(Y)QX - \eta(X)QY,$9, the quarter-symmetric metric connection $\eta$0 is given by

$\eta$1

with torsion $\eta$2 (Roy et al., 2021, V. et al., 2019). The induced curvature and Ricci tensors, and their traces, exhibit explicit deformations depending on both the contact structure and the induced metric.

A further generalization is present on $\eta$3-Sasakian manifolds via a two-parameter family of connections. The quarter-symmetric metric connection corresponds to the parameter choice $\eta$4, and the induced Ricci tensor retains symmetry. Ricci semi-symmetry for these connections implies the Einstein condition relative to $\eta$5 (Bahadır et al., 2018).

5. Submanifold Geometry and Classification Results

Quarter-symmetric metric connections significantly affect submanifold theory. For invariant submanifolds of $\eta$6-manifolds, the second fundamental form and the mean curvature vector with respect to the Levi-Civita and quarter-symmetric metric connections are identical (Hui et al., 2017, V. et al., 2019). Mininality, total umbilicity, and recurrence of submanifolds are equivalently characterized with respect to both connections.

For screen generic lightlike (SGL) submanifolds in indefinite Sasakian statistical manifolds, the induced connection, second fundamental form, and integrability of distributions involve the ambient quarter-symmetric connection and contact structure. Integrability, parallelism, and totally geodesic conditions on distributions are given explicitly in terms of the quarter-symmetric metric connection (Gupta et al., 14 Dec 2025).

6. Rigidity, Warped Products, and Global Classifications

Complete, simply connected Riemannian manifolds admitting a quarter-symmetric metric connection with parallel torsion are classified up to homothety. Such manifolds are isometric to warped products $\eta$7, where the torsion is expressed as $\eta$8. In these cases, the 1-form $\eta$9 is both closed and parallel, and the geometry is entirely determined by $Q$0 (Moroianu et al., 2020).

For Einstein warped product spaces admitting quarter-symmetric metric connections, if the total space is Einstein and the fiber has nonpositive scalar curvature, the warping function must be constant, reducing the product to a Riemannian product, ruling out nontrivial warped Einstein spaces with nonpositive fiber curvature and quarter-symmetric metric connection (Pal et al., 2022).

7. Applications, Inequalities, and Holonomy

In submanifold geometry, quarter-symmetric metric and Ricci quarter-symmetric metric connections lead to sharp scalar, Ricci, and Casorati-type curvature inequalities, with all equalities characterizing specific geometric situations such as totally umbilical or quasi-umbilical submanifolds (Wani et al., 2020). Further, connections of this type influence the behavior of Ricci-Yamabe solitons, conformal Killing fields, and projective pseudosymmetry conditions. Relations among canonical invariants (e.g., Weyl and holomorphically projective tensors) reveal additional geometric rigidity (Zlatanović et al., 2022).

Quarter-symmetric metric connections thus serve as a flexible and structurally rich generalization of Levi-Civita and semi-symmetric connections, encoding torsion via a precise 1-form–tensor mechanism. Their study illuminates a spectrum of modern problems in geometry, ranging from holonomy reduction to curvature inequalities, submanifold theory, and global manifold classification (Zlatanović et al., 2022, Pal et al., 2022, Moroianu et al., 2020, Wani et al., 2020, Hui et al., 2017, V. et al., 2019, Gupta et al., 14 Dec 2025, Roy et al., 2021, Bahadır et al., 2018).