Interpolating Sesqui-Harmonic Vector Fields

• The paper shows that interpolating sesqui-harmonic vector fields are critical points of an energy functional blending harmonic and biharmonic energies, yielding unified Euler–Lagrange equations.
• The methodology employs differential geometric tools such as horizontal–vertical decompositions and curvature-based operators to rigorously analyze these fields on various manifolds.
• Key findings indicate that the sign choices of delta parameters determine whether these fields are necessarily parallel or admit proper non-parallel solutions, influencing their analytic and geometric properties.

Interpolating sesqui-harmonic vector fields are special critical points of a family of variational functionals that interpolate between the harmonic and biharmonic vector field energies. Defined on a Riemannian or para-Kähler–Norden manifold $(M, g)$, these vector fields $X$ (or $\xi$) are regarded as maps from $(M,g)$ into the tangent bundle $TM$ equipped with either the Sasaki metric $g_S$ or a generalized %%%%6%%%%-Sasaki metric $g^\varphi$. The notion provides a unified variational framework for exploring first- and second-order geometric field theories, and the Euler–Lagrange equations naturally blend second- and fourth-order geometric operators, leading to a rich landscape of analytic and geometric properties (Kacimi et al., 2022, Zagane et al., 15 Jan 2026).

1. Mathematical Foundations and Definitions

Let $(M, g)$ be a Riemannian manifold of dimension $m$, and consider its tangent bundle $TM$. The Sasaki metric $g_S$ on $TM$ and its horizontal–vertical decomposition allow any smooth vector field $X \in \mathfrak{X}(M)$ to be viewed as a section (i.e., map) $X : (M, g) \to (TM, g_S)$ defined by $x \mapsto (x, X_x)$. For para-Kähler–Norden manifolds $(M^{2m}, \varphi, g)$, the $\varphi$-Sasaki metric $g^\varphi$ further adapts the construction by incorporating the structure tensor $\varphi$ as $g^\varphi(X^H, Y^H) = g(X, Y)$, $g^\varphi(X^V, Y^V) = g(X, \varphi Y)$, and $g^\varphi(X^H, Y^V) = 0$.

The tension field $\tau(X)$ of a vector field regarded as a map $X : M \to TM$ encapsulates its harmonicity, given by: $\tau(X) = (-S(X))^h + (-\bar\Delta X)^v,$ where

$$S(X) = \sum_{i=1}^m R(\nabla_{e_i} X, X) e_i,\qquad \bar\Delta X = -\operatorname{tr}_g (\nabla^2 X).$$

Here, $R$ is the curvature tensor of $(M, g)$, and $e_i$ is a local orthonormal frame. The bitension field is the second variation field relevant for biharmonicity: $$\tau_2(X) = \bar\Delta^X \tau(X) - \sum_{i=1}^m R^{TM}(\tau(X), dX(e_i)) dX(e_i),$$ where $R^{TM}$ is the curvature of the total space $(TM, g_S)$ or $(TM,g^\varphi)$ as appropriate.

2. The Interpolating Sesqui-Harmonic Functional

For $\alpha \in \mathbb{R}$, define the interpolating sesqui-harmonic energy as: $E_\alpha(X) = \alpha E_1(X) + (1 - \alpha) E_2(X),$ where

$$E_1(X) = \frac{1}{2} \int_M |dX|^2\, \operatorname{vol}_g,\qquad E_2(X) = \frac{1}{2} \int_M |\tau(X)|^2\, \operatorname{vol}_g.$$

For general constants $\delta_1, \delta_2 \in \mathbb{R}$, the formulation extends as $$E_{\delta_1,\delta_2}(X) = \delta_1 \int_M |dX|^2 + \delta_2 \int_M |\tau(X)|^2$$. The critical points of this energy (under variations of $X$) are defined as interpolating sesqui-harmonic vector fields.

The associated Euler–Lagrange expression, or interpolating sesqui-harmonic tension field, is

$$\tau_{\alpha}(X) = \alpha\, \tau(X) + (1-\alpha)\, \tau_2(X),$$

and $X$ is called an interpolating sesqui-harmonic map if $\tau_{\alpha}(X) = 0$. Restricting variational fields to vector field variations leads to a slightly weaker criticality condition (see Section 4).

3. Euler–Lagrange Equations and Characterization

Horizontal–Vertical Decomposition

The intricate field formula for the interpolating sesqui-harmonic tension decomposes into horizontal ($^h$) and vertical ($^v$) parts. For a section $X$ and parameters $\delta_1,\delta_2$, the Euler–Lagrange system (Kacimi et al., 2022, Zagane et al., 15 Jan 2026) reads: $$\begin{aligned} &\text{Horizontal:}\ &\quad -\delta_1\, S(X) - \delta_2\, \bar\Delta S(X) - \delta_2 R(X,\bar\Delta X)S(X) + \delta_2 \sum_{i=1}^m\Big\{R(X,\nabla_{e_i}\bar\Delta X)e_i \ &\hspace{2.5cm} - R(\nabla_{e_i} X,\bar\Delta X)e_i - R(e_i,S(X))e_i - (\nabla_{S(X)}R)(\nabla_{e_i}X,X)e_i \ &\hspace{2.5cm} + R(X,\nabla_{e_i}X)\nabla_{e_i}S(X) - R(X,R(e_i,S(X))X)e_i\Big\} = 0, \ &\text{Vertical:}\ &\quad -\delta_1\, \bar\Delta X - \delta_2\, \bar\Delta\bar\Delta X - \delta_2 \sum_{i=1}^m \big[\nabla_{e_i}R(e_i, S(X))X + R(e_i, \nabla_{e_i}S(X))X + 2R(e_i, S(X))\nabla_{e_i}X\big] = 0. \end{aligned}$$ An interpolating sesqui-harmonic map must satisfy both horizontal and vertical equations.

Vector Field Variations

When restricted to variations through vector fields, the Euler–Lagrange equation simplifies: $X$ is interpolating sesqui-harmonic if and only if

$$\delta_1\,\bar\Delta X + \delta_2\,\bar\Delta\bar\Delta X + \delta_2 \sum_{i=1}^m \big[(\nabla_{e_i}R)(e_i,S(X)) X + R(e_i, \nabla_{e_i}S(X)) X + 2R(e_i,S(X))\nabla_{e_i}X\big] = 0.$$

On $(M, g, \varphi)$ with $g^\varphi$ as in para-Kähler–Norden case, a similar structure holds (Zagane et al., 15 Jan 2026).

In the flat case ($R \equiv 0$), these reduce to the linear PDE: $$\delta_1\,\bar\Delta X + \delta_2\,\bar\Delta^2 X = 0,$$ which interpolates between vector field harmonicity ($\bar\Delta X=0$) and biharmonicity ($\bar\Delta^2 X=0$).

4. Key Theorems: Compact and Lie Group Cases

Compact, Oriented Manifolds

If $(M, g)$ is compact and oriented, with $\delta_1$ and $\delta_2$ of the same sign, any interpolating sesqui-harmonic vector field (or map) must be parallel: $\nabla X \equiv 0$ (Kacimi et al., 2022). This conclusion follows from a Bochner-type identity and vanishing of non-negative integrals arising in the first variation formula.

Homogeneous and Lie Group Backgrounds

Let $G$ be a Lie group with a left-invariant metric and $\Gamma \subset G$ a discrete subgroup making the quotient $\Gamma \backslash G$ compact. Left-invariant vector fields on $G$ descend naturally, and the same parallelism result holds: a left-invariant vector field is interpolating sesqui-harmonic if and only if it is parallel, assuming $\delta_1$ and $\delta_2$ have the same sign.

If instead $\delta_1$ and $\delta_2$ have opposite signs, non-parallel interpolating sesqui-harmonic vector fields arise. Explicit one-parameter families exist in the Heisenberg group (“Nil”), constructed via algebraic relations between frame components (Kacimi et al., 2022).

5. Extension to Para-Kähler–Norden and Pseudo-Riemannian Geometries

On para-Kähler–Norden manifolds $(M^{2m}, \varphi, g)$ with the $\varphi$-Sasaki metric $g^\varphi$, the general theory mirrors the Riemannian case with specific structural differences. The variation and Euler–Lagrange formulas are analogous, with additional dependence on $\varphi$-structure: $$\delta_1\,\bar\Delta\,\xi + \delta_2\,\bar\Delta^2\xi - \delta_2\,\operatorname{Tr}_g \Big[(\nabla_*R)(*,S(\xi))\xi + R(*, \nabla_* S(\xi))\xi + 2R(*, S(\xi))\nabla_*\xi\Big] = 0.$$ All harmonic vector fields (satisfying $S(\xi)=0$ and $\bar\Delta\xi=0$) are automatically interpolating sesqui-harmonic (Zagane et al., 15 Jan 2026).

6. Canonical Examples and Solution Structure

Explicit examples clarify the inclusions and intersections between harmonic, biharmonic, and interpolating sesqui-harmonic vector fields. For instance, on $\mathbb{R}^2$ with a flat para-Kähler–Norden structure, vector fields of the form $\xi = f(x) e_1$ satisfy ordinary differential equations depending on the parameter ratio $\lambda = \delta_1 / \delta_2$:

• Harmonic if $f'' - f' = 0$,
• Biharmonic if $f^{(4)} - 6f^{(3)} + 11f'' - 6f' = 0$,
• Interpolating sesqui-harmonic if

$$\delta_2(f^{(4)} - 6 f^{(3)} + 11 f'' - 6 f') - \delta_1 e^{2x} (f'' - f') = 0.$$

The general solution includes harmonic, biharmonic, and proper interpolating components. There also exist families of proper interpolating sesqui-harmonic vector fields that are neither harmonic nor biharmonic when both $\delta_1, \delta_2 \neq 0$ (Zagane et al., 15 Jan 2026).

7. Geometric Significance and Research Context

Interpolating sesqui-harmonic vector fields generalize both harmonic and biharmonic vector fields, creating a spectrum interpolated by the choice of $\alpha$ (or $\delta_1,\delta_2$). The theory connects to higher-order variational problems in both Riemannian and pseudo-Riemannian geometry, with implications for elasticity models and geometric flow constructions. The interplay between second- and fourth-order operators leads to novel rigidity phenomena on compact manifolds and flexibility in certain non-elliptic ranges, admitting proper solutions not reducible to classical cases (Kacimi et al., 2022, Zagane et al., 15 Jan 2026).

A plausible implication is that these results may further inform studies of geometric analysis on fiber bundles, submanifold theory, and the mathematical formulation of elasticity and field theories, particularly where both first- and second-order energies are physically or geometrically relevant. On homogeneous spaces, especially nilpotent or solvable cases, the explicit enumeration of interpolating sesqui-harmonic vector fields reveals subtle distinctions absent in strictly harmonic or biharmonic settings.