Einstein connection of nonsymmetric pseudo-Riemannian manifold

This paper extends Prvanović's coordinate-free formulation of the Einstein connection for almost Hermitian manifolds to almost contact metric manifolds satisfying an $f^2$-torsion condition, providing explicit torsion formulas and analyzing special cases within the Gray-Hervella classification.

The Gist

Imagine you are trying to understand the universe. For a long time, physicists like Albert Einstein had two separate rulebooks: one for gravity (how planets orbit and apples fall) and one for electromagnetism (how light and magnets work).

Einstein spent his later years trying to write a "Unified Field Theory"—a single, perfect rulebook that explains both forces at once. To do this, he proposed a strange idea: what if the fabric of space isn't just a smooth, symmetric sheet, but a slightly twisted, lopsided one?

This paper by Vladimir Rovenski and Milan Zlatanovi´c is like a mathematical repair manual for that twisted fabric. Here is the breakdown in simple terms:

1. The Twisted Fabric (The Nonsymmetric Manifold)

Think of the universe as a giant, stretchy trampoline.

• The Symmetric Part ($g$): This is the normal trampoline. It's smooth and even. This represents gravity.
• The Skewed Part ($F$): Now, imagine someone secretly twisted the trampoline so it has a slight spiral or a "kink" in it. This represents electromagnetism.

Einstein called this twisted trampoline a Nonsymmetric Riemannian Manifold. It's a fancy way of saying: "Space has two personalities: a smooth one and a twisted one."

2. The Problem: How to Walk on a Twisted Floor?

If you try to walk on a normal floor, you take straight steps. But if the floor is twisted, your steps get messed up. You might drift left or right without meaning to.

In math, this "drift" is called Torsion.

• The Levi-Civita Connection: This is the standard way of walking on a smooth floor (the rules of General Relativity).
• The Einstein Connection: This is a new set of walking rules designed specifically for the twisted floor. It tells you exactly how to step so that you stay on the path, accounting for the twist.

The authors' goal was to write down the exact instructions (formulas) for this "Einstein Connection" for various types of twisted floors.

3. The Special Rule: The "$f^2$-Torsion Condition"

The authors discovered that to make the math work, the twisted floor needs to follow a specific rhythm. They call this the $f^2$-torsion condition.

The Analogy:
Imagine a dancer spinning.

• If the dancer spins once ($f$), they turn.
• If they spin twice ($f^2$), they are back to facing the same way, but maybe upside down or shifted.
• The Condition says: "The way the floor twists when you spin twice must be consistent." It's like saying, "If I spin the floor twice, the twist I feel must be the same as if I just looked at the twist directly."

This condition is crucial because it simplifies the chaos. Without it, the math is a tangled mess. With it, the authors can solve the puzzle.

4. What Did They Actually Do?

The paper is divided into two main achievements:

A. Fixing the "Contact" Dance (Almost Contact Metric Manifolds)
Some spaces have a "center pole" (like a Reeb vector field, $\xi$) that everything spins around, like a carousel.

• The authors figured out how to calculate the walking rules (Einstein Connection) for these carousel-like spaces, provided they follow the "spin twice" rhythm.
• Result: They found that on these spaces, the "twist" (torsion) behaves very predictably. It turns out that if the space is "nice" enough, the twist is directly related to how the electromagnetic field ($F$) changes.

B. Fixing the "Weak" Dance (Weak Almost Hermitian Manifolds)
This is the big generalization. Previous mathematicians (like M. Prvanovi´c) solved this for "perfect" twisted floors (where the twist is exactly 90 degrees, like a standard circle).

• Rovenski and Zlatanovi´c asked: "What if the twist isn't perfect? What if it's 'weak' or 'squished'?"
• They created a master formula that works for any level of twist, as long as the "spin twice" rhythm holds.
• They introduced a new tool (a tensor called $\tilde{Q}$) which acts like a calibration dial. It measures how far the twist is from being perfect and adjusts the walking rules accordingly.

5. Why Does This Matter?

You might ask, "Who cares about twisted trampolines?"

• For Physicists: This helps refine Einstein's old Unified Field Theory. Even though Einstein's specific theory wasn't the final answer, the math of "nonsymmetric gravity" is still used today in advanced theories like String Theory and Quantum Gravity.
• For Mathematicians: They have now provided a "dictionary" to translate between the shape of space and the forces acting on it. They showed that for many complex shapes, the "twist" is actually just a reflection of how the electromagnetic field is changing.
• The "Gray-Hervella" Classes: The paper mentions these as a way to categorize different types of twisted floors (like sorting shoes by size and style). The authors showed that for many of these categories, the "Einstein Connection" is a "special" one, meaning the math simplifies beautifully.

Summary

Think of this paper as the instruction manual for navigating a universe that is slightly crooked.

1. Einstein said the universe is crooked (gravity + magnetism).
2. Previous mathematicians figured out how to walk on a perfectly crooked universe.
3. Rovenski and Zlatanovi´c figured out how to walk on a messily crooked universe, as long as the crookedness follows a specific rhythm (the $f^2$ condition).

They gave us the exact formulas to calculate the "drift" (torsion) so that we can understand how gravity and electromagnetism might dance together in a unified theory.

Technical Summary

Here is a detailed technical summary of the paper "Einstein connection of nonsymmetric Riemannian manifold" by Vladimir Rovenski and Milan Zlatanović.

1. Problem Statement

The paper addresses the problem of determining the explicit form of the Einstein connection ($\nabla$) on a nonsymmetric pseudo-Riemannian manifold (an NGT space).

• Context: In Einstein's Unified Field Theory (Nonsymmetric Gravitational Theory - NGT), the fundamental tensor is a nonsymmetric $(0,2)$-tensor $G = g + F$, where $g$ is a pseudo-Riemannian metric and $F$ is a non-zero skew-symmetric tensor (associated with electromagnetism).
• The Connection: The Einstein connection $\nabla$ is a linear connection with torsion $T$ satisfying the metricity condition:
  $$(\nabla_X G)(Y, Z) = -G(T(X, Y), Z)$$
• The Gap: While M. Prvanović (1995) derived the explicit form of this connection for almost Hermitian manifolds (where $f^2 = -I$), the general case for manifolds where $f^2 \neq -I$ (specifically weak almost Hermitian manifolds and almost contact metric manifolds) remained unresolved without additional constraints.
• Objective: To generalize Prvanović's result to a broader class of manifolds by introducing a specific constraint, the $f^2$-torsion condition, and deriving explicit coordinate-free formulas for the torsion tensor $T$ and the connection $\nabla$.

2. Methodology

The authors employ differential geometry techniques, specifically utilizing tensor calculus on manifolds equipped with a $(1,1)$-tensor field $f$ and a metric $g$.

• Decomposition: They decompose the fundamental tensor $G$ into symmetric ($g$) and skew-symmetric ($F$) parts, defining $F(X, Y) = g(X, fY)$.
• The $f^2$-Torsion Condition: The core methodological innovation is the imposition of the condition:
  $$T(f^2 X, Y) = T(X, f^2 Y) = f^2 T(X, Y)$$
  This condition generalizes the behavior of torsion in classical almost Hermitian geometry (where $f^2 = -I$) to cases where $f^2$ is not necessarily $-I$ (e.g., weak almost Hermitian or almost contact structures).
• Difference Tensor: They utilize the difference tensor $K$ between the Einstein connection $\nabla$ and the Levi-Civita connection $\nabla^g$, defined by $K_{XY} = \nabla_X Y - \nabla^g_X Y$.
• Algebraic Manipulation: The authors derive a system of linear equations involving the torsion $T$, the covariant derivative of the fundamental form $\nabla^g F$, the exterior derivative $dF$, and the tensor $\tilde{Q} = -f^2 - I$. By applying cyclic sums and utilizing the $f^2$-torsion condition, they isolate $T$ in terms of known geometric quantities.
• Case Analysis: The derivation is split into specific cases:
  1. Almost Contact Metric Manifolds: Where $f^2 = -I + \eta \otimes \xi$.
  2. Weak Almost Hermitian Manifolds: Where $f$ is non-singular but $f^2 \neq -I$.
  3. Special Cases: Totally skew-symmetric torsion and "special" Einstein connections (where $K$ is skew-symmetric in the first two arguments).

3. Key Contributions and Results

A. Generalization of Prvanović's Theorem

The paper successfully extends Prvanović's 1995 result.

• Theorem 2.8 (Recap): For almost Hermitian manifolds ($f=J, J^2=-I$), the torsion is given by a specific formula involving $\nabla^g F$ and $dF$.
• Theorem 4.3 (Main Result): For a general nonsymmetric pseudo-Riemannian space satisfying the $f^2$-torsion condition, the authors derive a complex but explicit formula (Eq. 4.4) for the torsion $T$. This formula expresses $T$ in terms of:
  • $\nabla^g F$ (covariant derivative of the fundamental form).
  • $dF$ (exterior derivative).
  • The tensor $\tilde{Q} = -f^2 - I$.
  • The inverse of the tensor $P = I - f^2$ (assuming non-degeneracy).
• Reduction: The authors demonstrate that when $f^2 = -I$ (the almost Hermitian case), their general formula reduces exactly to Prvanović's coordinate-free solution.

B. Almost Contact Metric Manifolds

• Theorem 3.3: For an almost contact metric manifold $(M, f, \xi, \eta, g)$ satisfying the $f^2$-torsion condition, the authors prove that:
  • $\nabla^g \xi = 0$ and $\nabla^g \eta = 0$ (the Reeb vector field is parallel).
  • The torsion $T$ vanishes when any argument is $\xi$ (i.e., $T$ is horizontal).
  • The torsion on the horizontal distribution follows the same structural formula as the almost Hermitian case (Eq. 3.8).
• Corollary 3.7 & 3.8: They provide explicit formulas for "special" Einstein connections (where the difference tensor is skew-symmetric in the first two indices) and conditions under which the torsion is totally skew-symmetric (leading to the "almost-nearly cosymplectic" case).

C. Weak Almost Hermitian Manifolds

• Corollary 4.5: The paper applies the general theorem to weak almost Hermitian manifolds (where $f$ is non-singular but $f^2 \neq -I$).
• Example 4.4 (Weighted Product): The authors construct a specific example using a Riemannian product of almost Hermitian manifolds with different scaling factors ($\lambda_j$). They show how the torsion splits according to the product structure and provide a simplified formula for the torsion component on each factor. This illustrates how the Einstein connection behaves on "weak" structures that are not standard almost Hermitian.

D. Classification and Symmetry

• The paper discusses the Gray-Hervella classes ($W_1, W_3, W_4$, etc.) of almost Hermitian manifolds.
• They identify specific classes where the Einstein connection becomes "special" (satisfying $K_{XY} = -K_{YX}$), linking the algebraic properties of the torsion to the geometric classification of the manifold.

4. Significance

• Unification of Theories: The work bridges the gap between classical almost Hermitian geometry and the more general "weak" metric structures, providing a unified framework for Einstein's nonsymmetric gravity in these contexts.
• Explicit Formulas: Prior to this work, explicit formulas for the Einstein connection in the presence of non-standard $f^2$ structures were not available. The derived formulas (Eq. 4.4) allow for the direct computation of the connection and torsion given the metric and the tensor $f$.
• New Geometric Tools: The introduction of the $f^2$-torsion condition serves as a powerful constraint that simplifies the otherwise intractable system of equations for the Einstein connection in nonsymmetric geometry.
• Applications in Physics: By providing explicit solutions for NGT spaces, the paper offers potential tools for modeling gravitational theories that incorporate electromagnetic fields via the skew-symmetric tensor $F$, particularly in scenarios involving weak structures or contact geometry (relevant in string theory and supersymmetric models).
• Future Directions: The authors lay the groundwork for studying weak (para-)metric $f$-manifolds and connections with totally skew-symmetric torsion, suggesting a path for further research in modified gravity theories.

In summary, Rovenski and Zlatanović have successfully generalized a classic result in differential geometry to a much broader class of manifolds, providing the necessary mathematical machinery to analyze Einstein connections in nonsymmetric gravitational theories beyond the standard almost Hermitian setting.