Concircularly Semi-Symmetric Metric Connection

• Concircularly semi-symmetric metric connection is a torsionful metric connection where the generator's one-form satisfies a concircular condition, leading to distinct curvature tensors.
• Its curvature hierarchy links the Levi-Civita curvature to derived tensors, unveiling conditions for Einstein, Ricci pseudo-symmetry, and perfect fluid space-time formations.
• In Lorentzian manifolds with a unit timelike generator, the connection reduces to a semi-symmetric metric P-connection, characterizing generalized Robertson–Walker space-times with significant relativistic applications.

A concircularly semi-symmetric metric connection is a torsionful metric connection obtained by imposing a concircular condition on the $1$-form that generates a semi-symmetric metric connection. In the Lorentzian framework developed in “Lorentzian manifolds equipped with a concircularly semi-symmetric metric connection” (Maksimović et al., 3 May 2025), this structure is not merely a formal variant of the Levi-Civita connection: under a unit timelike hypothesis on the generator, it collapses to a semi-symmetric metric $P$-connection and forces the underlying Lorentzian manifold to be a generalized Robertson–Walker (GRW) space-time. The theory then branches into a curvature hierarchy involving several associated curvature tensors, rigidity criteria for Einstein geometry, Ricci pseudo-symmetry phenomena, and applications to perfect fluid space-times. Later pseudo-Riemannian work develops parallel Einstein-type and quasi-Einstein consequences for related formulations of the same notion (Maksimović et al., 1 Oct 2025).

1. Defining structure

Let $(M,g)$ be a pseudo-Riemannian manifold of dimension $n>2$, with Levi-Civita connection $\nabla$. A semi-symmetric metric connection $V$ is defined by

$V_XY=\nabla_XY+T(Y)X-g(X,Y)P,$

where $T$ is a $1$-form and $P$ is its associated vector field, so that

$P$0

Its torsion tensor is

$P$1

The connection is metric, $P$2, but it is not torsion-free. The same framework also admits the associated symmetric and dual connections

$P$3

and

$P$4

although the primary object is $P$5 itself (Maksimović et al., 3 May 2025).

The connection is called concircularly semi-symmetric metric when the generating $P$6-form satisfies

$P$7

for some scalar function $P$8. A fundamental derived identity is

$P$9

This has several immediate consequences: $(M,g)$0 is concircular in Fialkow’s sense with respect to $(M,g)$1, it is torse-forming relative to the connection, and if $(M,g)$2, then $(M,g)$3, so $(M,g)$4 is parallel with respect to $(M,g)$5. In particular, the connection reduces to a semi-symmetric metric $(M,g)$6-connection precisely when

$(M,g)$7

This criterion isolates the exact transition from the concircularly constrained semi-symmetric metric connection to the parallel-generator case (Maksimović et al., 3 May 2025).

A later pseudo-Riemannian formulation uses the same semi-symmetric metric ansatz

$(M,g)$8

with $(M,g)$9, and imposes

$n>2$0

In that treatment, one again obtains

$n>2$1

so the parallel-generator condition is

$n>2$2

(Maksimović et al., 1 Oct 2025). The notation differs, but the same structural motif persists: a semi-symmetric metric connection is constrained by a concircular-type equation on its generator.

2. Curvature hierarchy

A distinctive feature of the theory is the use of several curvature tensors attached to the connection. The generalized curvature action on a $n>2$3-tensor $n>2$4 is

$n>2$5

and the Tachibana tensor is

$n>2$6

where

$n>2$7

These operators are used to compare the Levi-Civita curvature tensor $n>2$8 with six curvature tensors $n>2$9 associated with the non-symmetric connection (Maksimović et al., 3 May 2025).

Representative identities include

$\nabla$0

$\nabla$1

and

$\nabla$2

Their contractions yield corresponding Ricci tensors, including

$\nabla$3

$\nabla$4

$\nabla$5

and

$\nabla$6

The scalar curvatures satisfy analogous formulas (Maksimović et al., 3 May 2025).

This curvature hierarchy is not vacuous in the GRW regime. The curvature tensors $\nabla$7 are non-zero, and the corresponding Ricci tensors are also non-zero. That nonvanishing result is methodologically important: it excludes trivial flatness and redirects the analysis toward weaker curvature restrictions such as semi-symmetry and pseudo-symmetry (Maksimović et al., 3 May 2025).

In a later pseudo-Riemannian treatment, six Ricci tensors and their scalar curvatures are again computed for the concircularly semi-symmetric metric connection, and a notable point is that all of these Ricci tensors are symmetric despite the underlying connection being non-symmetric. This symmetry is then used to define Einstein-type manifolds of several kinds (Maksimović et al., 1 Oct 2025).

3. Lorentzian reduction to GRW space-times

The most rigid result arises in the Lorentzian case. Assume $\nabla$8 is Lorentzian and the generator $\nabla$9 is unit timelike: $V$0 Differentiating this relation and using the concircular condition gives

$V$1

hence

$V$2

Substituting into the basic identity for $V$3 yields

$V$4

Therefore $V$5 is parallel with respect to $V$6, and the connection becomes a semi-symmetric metric $V$7-connection. This is the content of Theorem 3.1 in (Maksimović et al., 3 May 2025).

The same paper then invokes a known characterization: an $V$8-dimensional Lorentzian manifold $V$9 equipped with a semi-symmetric metric $V_XY=\nabla_XY+T(Y)X-g(X,Y)P,$0-connection whose associated vector $V_XY=\nabla_XY+T(Y)X-g(X,Y)P,$1 is a unit timelike torse-forming vector field is a GRW space-time. Combining that characterization with the previous theorem yields the corollary that an $V_XY=\nabla_XY+T(Y)X-g(X,Y)P,$2-dimensional Lorentzian manifold $V_XY=\nabla_XY+T(Y)X-g(X,Y)P,$3, $V_XY=\nabla_XY+T(Y)X-g(X,Y)P,$4, equipped with a concircularly semi-symmetric metric connection whose associated vector $V_XY=\nabla_XY+T(Y)X-g(X,Y)P,$5 is unit timelike is a GRW space-time (Maksimović et al., 3 May 2025).

A later formulation expresses the same Lorentzian reduction through the Levi-Civita derivative. If $V_XY=\nabla_XY+T(Y)X-g(X,Y)P,$6, then the concircular condition forces

$V_XY=\nabla_XY+T(Y)X-g(X,Y)P,$7

equivalently,

$V_XY=\nabla_XY+T(Y)X-g(X,Y)P,$8

Thus $V_XY=\nabla_XY+T(Y)X-g(X,Y)P,$9 is a unit timelike torse-forming vector field, the connection is again a semi-symmetric metric $T$0-connection, and the Lorentzian manifold is a GRW space-time (Maksimović et al., 1 Oct 2025).

4. Einstein geometry, Ricci semi-symmetry, and pseudo-symmetry

Once the GRW reduction is in place, the principal symmetry questions concern the action of curvature on the Ricci tensor. A central theorem states that a GRW space-time $T$1 is Einstein if and only if

$T$2

The proof uses the special identities

$T$3

and concludes that

$T$4

which is the Einstein condition in the normalization adopted there (Maksimović et al., 3 May 2025).

A second equivalent characterization is

$T$5

Thus Einstein geometry can be detected either by annihilation conditions involving $T$6 and $T$7, or by an equality between the curvature actions of $T$8 and the Levi-Civita curvature tensor (Maksimović et al., 3 May 2025).

The relation between Ricci semi-symmetry and Einstein geometry is especially sharp. In this setting, a GRW space-time is Ricci semi-symmetric if and only if it is Einstein. Here Ricci semi-symmetry is therefore not merely a weaker condition. It is equivalent to the Einstein condition for the class under consideration (Maksimović et al., 3 May 2025).

The same work also derives a constant-type pseudo-symmetry statement. In the semi-symmetric metric $T$9-connection setting,

$1$0

is equivalent to the manifold being Ricci pseudo-symmetric of constant type, with defining relation

$1$1

This fits into the broader network of curvature conditions

$1$2

whose interactions in higher dimensions are a stated motivation for the theory (Maksimović et al., 3 May 2025).

5. Perfect fluid space-times and relativistic consequences

In the relativistic part of the theory, a Lorentzian manifold is a perfect fluid space-time if its Ricci tensor has the form

$1$3

Within the GRW and semi-symmetric metric $1$4-connection setting, this quasi-Einstein-type ansatz becomes highly constrained. One theorem states that a perfect fluid space-time $1$5 equipped with a semi-symmetric metric $1$6-connection is Ricci pseudo-symmetric of constant type and satisfies

$1$7

A further corollary gives the stronger restriction

$1$8

The paper also states that perfect fluid space-times of the “$1$9-th kind” coincide with perfect fluid space-times in this GRW/semi-symmetric $P$0-connection framework (Maksimović et al., 3 May 2025).

The final physical application in that Lorentzian study concerns Einstein’s field equations without cosmological constant,

$P$1

for a perfect fluid stress tensor

$P$2

where $P$3 is the energy density and $P$4 the isotropic pressure. The resulting theorem states that, in a perfect fluid space-time satisfying these equations, the strong energy condition is violated. The key inequality is

$P$5

In the terminology of the paper, the required strong energy condition inequalities therefore fail (Maksimović et al., 3 May 2025).

A later application with cosmological constant reaches a different but related endpoint. There the stress-energy tensor is written

$P$6

and the conservation law, together with the torse-forming relation of the unit timelike generator, forces

$P$7

Hence

$P$8

described there as the phantom barrier value for dark energy. That later result concerns Einstein’s field equations with cosmological constant and complements, rather than duplicates, the earlier strong-energy-condition violation without cosmological constant (Maksimović et al., 1 Oct 2025).

6. Terminology and related constructions in the literature

The label “concircularly semi-symmetric metric connection” is not uniform across adjacent literatures. In the Lorentzian work (Maksimović et al., 3 May 2025), it denotes a semi-symmetric metric connection whose generator satisfies a specific concircular equation. By contrast, “Index of quasi-conformally symmetric semi-Riemannian manifolds” (Tripathi et al., 2012) studies $P$9-concircularly symmetric manifolds, defined by

$P$00

where $P$01 is the concircular curvature tensor of a metric connection $P$02. That paper explicitly notes that it does not separately define “$P$03-concircularly semi-symmetric” in the standard curvature-operator sense. Its main concircular theorem is a rigidity statement: $P$04 meaning that the metric is, up to scalar multiple, the only $P$05-parallel symmetric $P$06-tensor (Tripathi et al., 2012).

A neighboring but distinct notion appears on Weyl manifolds. In “A Necessary and Sufficient Condition on the Weyl Manifolds Admitting a Semi Symmetric Non-Metric Connection to be S-Concircular” (Bastan, 2014), the relevant condition is S-concircularity for a semi-symmetric non-metric connection: $P$07 The necessary and sufficient condition is that the concircular curvature tensors of the symmetric connection and the semi-symmetric non-metric connection coincide,

$P$08

This is an equality of concircular curvature tensors, not the Lorentzian generator condition used in (Maksimović et al., 3 May 2025).

Related metric-connection constructions also occur in contact and curve geometry. On Kenmotsu manifolds, a generalized symmetric metric connection

$P$09

specializes to the semi-symmetric metric connection when $P$10, and concircular curvature invariance forces a generalized $P$11-Einstein Ricci form (Alghamdi et al., 2018). In three-dimensional curve theory, semi-symmetric metric connections modify the Frenet equations and thereby alter the notions of geodesic, circle, and helix, although that work does not develop concircularly semi-symmetric metric connections directly (Güvenç, 2024).

A plausible implication of this literature pattern is that the modern usage of the term has become increasingly connection-centered: rather than starting from the operator condition $P$12, recent work emphasizes a semi-symmetric metric connection whose generator satisfies a concircular or torse-forming constraint, from which curvature, Einstein-type, and relativistic consequences are then deduced (Maksimović et al., 3 May 2025).