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 $\varphi_1, \varphi_2, \varphi_3: TM \to (0,\infty)$: $(M,g)$0 where $(M,g)$1 are the horizontal lifts and $(M,g)$2 the vertical lifts of tangent vectors in $(M,g)$3. The metric is Riemannian provided the $(M,g)$4 matrix with entries $(M,g)$5 is positive-definite at every point and $(M,g)$6 is symmetric in its arguments. The "standard" $(M,g)$7-Sasaki metric on a para-Kähler–Norden manifold $(M,g)$8 is characterized by:

• $(M,g)$9,
• $\varphi$0,
• $\varphi$1 (Zagane, 2023, Zagane et al., 15 Jan 2026, Vang et al., 2018).

On Infinite-Dimensional Spaces:

In the setting of Sasaki manifolds $\varphi$2, the $\varphi$3-Sasaki metric refers to the analogue of the Mabuchi $\varphi$4 metric on the infinite-dimensional space of Sasaki potentials

$\varphi$5

with the Riemannian pairing

$\varphi$6

(Franzinetti, 2020).

Finite-Dimensional Model for Trajectory Spaces:

For manifold-valued cubic Bézier spline trajectories, the $\varphi$7-Sasaki metric is realized as the pullback of the product Sasaki metric on $\varphi$8 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 $\varphi$9 admits a block decomposition by horizontal and vertical lifts, with connection coefficients expressed as a combination of the base connection $(M,g)$0, the curvature tensor $(M,g)$1, and derivatives of the scalar weight functions $(M,g)$2. For weights independent of fiber coordinates and with $(M,g)$3, the connection reduces to the standard Sasaki case:

• Horizontal–horizontal: $(M,g)$4
• Horizontal–vertical: involves $(M,g)$5, the curvature, and weight functions
• Vertical–vertical: depends only on vertical gradients of weight functions

For para-Kähler–Norden base $(M,g)$6, the connection is explicitly: $(M,g)$7 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 $(M,g)$8 of $(M,g)$9 retains a block structure; the horizontal–horizontal component encodes the base curvature $\varphi: TM \to TM$0 plus terms involving derivatives of $\varphi: TM \to TM$1 and interactions with the vertical structure. Sectional curvatures and scalar curvature can be computed by standard contraction, with explicit dependence on $\varphi: TM \to TM$2 and the base metric. For para-Kähler–Norden settings, the presence of the $\varphi: TM \to TM$3-tensor alters the vertical–vertical metric and couples curvature via $\varphi: TM \to TM$4 (Zagane et al., 15 Jan 2026, Zagane, 2023).

3. Geodesics and Variational Principles

Geodesics:

Geodesic flow on $\varphi: TM \to TM$5 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: TM \to TM$6-Sasaki tangent bundle of a para-Kähler–Norden manifold:

• Horizontal component: $\varphi: TM \to TM$7
• Vertical component: $\varphi: TM \to TM$8 (Zagane, 2023, Zagane et al., 15 Jan 2026).

Variational Frameworks:

For sections $\varphi: TM \to TM$9, energy and bienergy functionals induce Euler-Lagrange equations involving rough Laplacians and curvature-coupling terms specific to the $\varphi$0-Sasaki metric. The tension field admits a horizontal/vertical decomposition, with

$\varphi$1

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$2	Key Features
Para-Kähler–Norden manifold	Para-complex structure, $\varphi$3	Modifies vertical component, curvature coupling
Sasaki manifold potential space	Basic function $\varphi$4, $\varphi$5	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$6-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$7-Sasaki setting yields new classes of harmonic, biharmonic, and sesqui-harmonic vector fields, where the vertical component is altered by $\varphi$8-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 ($\varphi$9): All curvature expressions vanish; the only nontrivial contributions to curvature and connection of $g_\varphi$0 arise from the fiber-dependent weight functions $g_\varphi$1. When the weights depend only on $g_\varphi$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 $g_\varphi$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 $g_\varphi$4 (Vang et al., 2018).
• Para-Kähler–Norden setting with explicit $g_\varphi$5: Eg: $g_\varphi$6 with $g_\varphi$7 leads to simple, decoupled geodesic equations for the fiber, showing the natural adaptation of the $g_\varphi$8-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 $g_\varphi$9-Sasaki metric space (completion of $TM$0) 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 $TM$1-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 $TM$2-unit tangent bundle $TM$3, 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 $TM$4 (Zagane, 2023).

7. Research Directions and Significance

The $TM$5-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 $TM$6-Sasaki construction, allowing coupling between horizontal and vertical bundles, fiberwise modification by $TM$7, and adaptability to both finite and infinite-dimensional settings, continues to make it a central tool in both theoretical and applied differential geometry.