Varphi-Sasaki Metric Overview

• Varphi-Sasaki metric is a generalization of the classical Sasaki metric that introduces a tensor field to modify horizontal and vertical structures on tangent bundles.
• It incorporates finite- and infinite-dimensional models with explicit formulas for connections, geodesics, and curvature, enabling detailed geometric analysis.
• Applications include harmonic analysis, variational problems, and statistical computations on manifold-valued trajectory spaces, offering practical insights in Riemannian geometry.

A $\varphi$-Sasaki metric is a generalization of the classical Sasaki metric, defined on the tangent bundle or spaces canonically associated to a base manifold $(M,g)$ endowed with additional geometric structure, typically a tensor field $\varphi$. Such metrics play a fundamental role in the differential geometry of tangent bundles, spaces of potentials on Sasaki manifolds, and in constructing computable Riemannian geometries on manifold-valued path spaces. Several variants are in use, including finite-dimensional concrete models, infinite-dimensional analogues in functional spaces, and versions tailored to para-Kähler–Norden and related geometries.

1. Formal Definitions

Canonical Form on Tangent Bundles:

Let $(M,g)$ be a Riemannian manifold and $\varphi: TM \to TM$ a (1,1)-tensor field (commonly, the canonical product structure or a para-complex structure). The general $\varphi$-Sasaki metric $g_\varphi$ on $TM$ is defined at each $(p,u) \in TM$ by scalar-valued functions $\varphi_1, \varphi_2, \varphi_3: TM \to (0,\infty)$: $$g_\varphi\bigl(X^h+Y^v, U^h+V^v\bigr)_{(p,u)} = \varphi_1\,g_p(X,U) + \varphi_2\,g_p(Y,V) + \varphi_3\,g_p(X,V) + \varphi_3\,g_p(Y,U)$$ where $X^h, U^h$ are the horizontal lifts and $Y^v, V^v$ the vertical lifts of tangent vectors in $T_pM$. The metric is Riemannian provided the $2\times 2$ matrix with entries $\varphi_1, \varphi_2, \varphi_3$ is positive-definite at every point and $\varphi_3$ is symmetric in its arguments. The "standard" $\varphi$-Sasaki metric on a para-Kähler–Norden manifold $(M, \varphi, g)$ is characterized by:

• $g^\varphi(X^h, Y^h) = g(X,Y)$,
• $g^\varphi(X^h, Y^v) = 0$,
• $g^\varphi(X^v, Y^v) = g(\varphi X, \varphi Y)$ (Zagane, 2023, Zagane et al., 15 Jan 2026, Vang et al., 2018).

On Infinite-Dimensional Spaces:

In the setting of Sasaki manifolds $(M, \xi, \eta, \Phi, g)$, the $\varphi$-Sasaki metric refers to the analogue of the Mabuchi $L^2$ metric on the infinite-dimensional space of Sasaki potentials

$$\mathcal{H}(M,\xi,d\eta) = \{\phi\in C^\infty_B(M): d\eta + dd^c\phi > 0\}$$

with the Riemannian pairing

$$(\psi_1, \psi_2)_\phi = \int_M \psi_1 \psi_2\, \eta \wedge (d\eta + dd^c\phi)^n$$

(Franzinetti, 2020).

Finite-Dimensional Model for Trajectory Spaces:

For manifold-valued cubic Bézier spline trajectories, the $\varphi$-Sasaki metric is realized as the pullback of the product Sasaki metric on $(TU)^{L+1}$ via a control-point diffeomorphism, yielding explicit closed-form formulas for Riemannian computations (Nava-Yazdani et al., 2023).

2. Levi-Civita Connection and Curvature

Connection Structure:

The Levi-Civita connection for $(TM, g_\varphi)$ admits a block decomposition by horizontal and vertical lifts, with connection coefficients expressed as a combination of the base connection $\nabla$, the curvature tensor $R$, and derivatives of the scalar weight functions $\varphi_i$. For weights independent of fiber coordinates and with $\varphi_3 = 0$, the connection reduces to the standard Sasaki case:

• Horizontal–horizontal: $$\nabla_{X^h}^{\varphi}Y^h = (\nabla_X Y)^h + \text{curvature/weight terms}$$
• Horizontal–vertical: involves $\nabla_X Y$, the curvature, and weight functions
• Vertical–vertical: depends only on vertical gradients of weight functions

For para-Kähler–Norden base $(M^{2m}, \varphi, g)$, the connection is explicitly: $$\widetilde{\nabla}_{X^h} Y^h = (\nabla_X Y)^h - (R(X,Y)\xi)^v,\quad \widetilde{\nabla}_{X^h}Y^v = (\nabla_X Y)^v + (R(X,\xi)Y)^h$$ with vertical–vertical and mixed terms as given in (Zagane, 2023, Vang et al., 2018, Zagane et al., 15 Jan 2026).

Curvature Properties:

The Riemann curvature $\widetilde{R}$ of $g_\varphi$ retains a block structure; the horizontal–horizontal component encodes the base curvature $R$ plus terms involving derivatives of $\varphi$ and interactions with the vertical structure. Sectional curvatures and scalar curvature can be computed by standard contraction, with explicit dependence on $\varphi_i$ and the base metric. For para-Kähler–Norden settings, the presence of the $\varphi$-tensor alters the vertical–vertical metric and couples curvature via $g(\varphi X, \varphi Y)$ (Zagane et al., 15 Jan 2026, Zagane, 2023).

3. Geodesics and Variational Principles

Geodesics:

Geodesic flow on $(TM, g_\varphi)$ is governed by coupled ODEs whose horizontal and vertical components involve both base geodesic acceleration and curvature terms modulated by the connection coefficients. On the $\varphi$-Sasaki tangent bundle of a para-Kähler–Norden manifold:

• Horizontal component: $\nabla_{\dot y}\dot y - R(\dot\xi,\xi)\dot y = 0$
• Vertical component: $\nabla_{\dot y}\dot\xi + R(\dot\xi,\xi)\xi = 0$ (Zagane, 2023, Zagane et al., 15 Jan 2026).

Variational Frameworks:

For sections $\xi:M\to TM$, energy and bienergy functionals induce Euler-Lagrange equations involving rough Laplacians and curvature-coupling terms specific to the $\varphi$-Sasaki metric. The tension field admits a horizontal/vertical decomposition, with

$$\tau(\xi) = HS(\xi) - V(\bar{\Delta}\xi),\quad S(\xi) = \operatorname{Tr}_g \{ R(\varphi \xi, \nabla_{(\cdot)}\xi)\cdot \}$$

leading to higher-order PDEs for biharmonicity and sesqui-harmonicity (Zagane et al., 15 Jan 2026).

4. Special Models and Applications

Setting	Role of $\varphi$	Key Features
Para-Kähler–Norden manifold	Para-complex structure, $\varphi^2=I$	Modifies vertical component, curvature coupling
Sasaki manifold potential space	Basic function $\phi$, $d\eta+dd^c\phi>0$	Infinite-dimensional CAT(0) structure, geodesic convexity
Bézier spline trajectory space	Pullback via Bézier diffeomorphism	Explicit finite-dimensional metric, enables Riemannian statistics

• In Riemannian statistics and geometric data analysis, the $\varphi$-Sasaki metric enables explicit calculation of distances, geodesics, and Fréchet means on spaces such as spline trajectories, bypassing the need for infinite-dimensional integration and reducing to finite-dimensional problems with closed-form expressions (Nava-Yazdani et al., 2023).
• For harmonic analysis in pseudo-Riemannian geometry, the $\varphi$-Sasaki setting yields new classes of harmonic, biharmonic, and sesqui-harmonic vector fields, where the vertical component is altered by $\varphi$-dependent terms, leading to genuine higher-order (fourth-order) criticality conditions (Zagane et al., 15 Jan 2026).

5. Examples and Explicit Computations

• Flat Euclidean base ($M=\mathbb{R}^n$): All curvature expressions vanish; the only nontrivial contributions to curvature and connection of $g_\varphi$ arise from the fiber-dependent weight functions $\varphi_i$. When the weights depend only on $\|u\|^2$ in the fiber, geodesic equations decouple into base and fiber components. This allows simple analytic solvability (Vang et al., 2018, Zagane, 2023).
• Compact Lie group $M=SO(3)$: Nonabelian structure introduces explicit computations in terms of Lie brackets and their action on left-invariant frames. The vertical-horizontal coupling and curvature terms can be written in terms of structure constants and derivatives of the scalar weights $\varphi_i$ (Vang et al., 2018).
• Para-Kähler–Norden setting with explicit $\varphi$: Eg: $M = \mathbb{R}^2$ with $\varphi = \operatorname{diag}(1,-1)$ leads to simple, decoupled geodesic equations for the fiber, showing the natural adaptation of the $\varphi$-Sasaki metric to product or para-complex geometries (Zagane, 2023).

6. Geometric and Analytical Properties

• CAT(0) and Curvature Bounds: In the infinite-dimensional setting of Sasaki potentials, the $\varphi$-Sasaki metric space (completion of $\mathcal{H}$) is CAT(0), with non-positive Alexandrov curvature, guaranteeing uniqueness of minimal geodesics and convexity of energy functionals. This mirrors analogous properties in Kähler geometry, providing a crucial tool for existence and uniqueness in problems of Sasaki–Einstein and cscS metrics (Franzinetti, 2020).
• Versatility: The metric structure is robust: for product as well as pullback constructions (such as in spline trajectory spaces), the $\varphi$-Sasaki metric transforms infinite-dimensional geometric computations to tractable, low-dimensional scenarios amenable to explicit Riemannian calculus (Nava-Yazdani et al., 2023).
• Constraint Adaptations: On the $\varphi$-unit tangent bundle $T_1^\varphi M$, the metric and connection are restricted to the unit sphere in the fiber, involving additional adjustment terms. For geodesic constraints, vertical equations include Lagrange terms enforcing $g(\xi,\xi)=1$ (Zagane, 2023).

7. Research Directions and Significance

The $\varphi$-Sasaki metric family underpins a range of current research, including:

• Higher-order harmonicity problems (biharmonic and interpolating sesqui-harmonic vector fields) in pseudo- and paracomplex geometries, leading to new PDE systems for variational field theories (Zagane et al., 15 Jan 2026).
• Foundations of metric geometry in infinite-dimensional settings related to geometric analysis and complex geometry, especially in the study of spaces of potentials with CAT(0) structure and applications to extremal Sasaki metrics (Franzinetti, 2020).
• Statistical and computational geometry of manifold-valued trajectory data, especially for practical tasks such as regression and mean computation on spaces of splines, with applications in data science (e.g., hurricane track analysis) (Nava-Yazdani et al., 2023).

The flexibility of the $\varphi$-Sasaki construction, allowing coupling between horizontal and vertical bundles, fiberwise modification by $\varphi$, and adaptability to both finite and infinite-dimensional settings, continues to make it a central tool in both theoretical and applied differential geometry.