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Non-Natural Metrics on the Tangent Bundle

Natural metrics provide a way to induce a metric on the tangent bundle from the metric on its base manifold. The most studied type is the Sasaki metric, which applies the base metric separately to the vertical and horizontal components. We study a more general class of metrics which introduces interactions between the vertical and horizontal components, with scalar weights. Additionally, we explicitly clarify how to apply our and other induced metrics on the tangent bundle to vector fields where the vertical component is not constant along the fibers. We give application to the Special Orthogonal Group SO(3) as an example.

1. Introduction. The study of tangent bundles and their relationship to the base manifold often rely on the Sasaki metric. However, we may gain valuable mathematical and physical insights by choosing a more general metric. For mechanical systems, tangent bundles arise naturally where the manifold is the configuration space and the Lagrangian mechanics involve the configurations and their velocities as state variables [1]. A fundamental process is damping, in which the changes on the configuration depend on changes to its velocity. Hence, we want to study metrics where this kind of interaction is considered. The Sasaki metric does not consider these kind of interactions.
The contributions of this paper are the generalization of the Sasaki metric and the derivation of the corresponding Levi-Civita connection on the tangent bundle. We also clarify the application of the results to general vector fields that are not constant along fibers.
The paper is organized as follows. A brief overview of relevant differential geometry concepts are provided in section 2, our main results are in section 3 and section 4, examples are in section 5, and the conclusions follow in section 6.
2. Background. Let M be a n-dimensional differentiable manifold equipped with a Riemannian metric g and T M the tangent bundle of M . For a point p ∈ M , let T p M denote the tangent space of M at p. A pointP ∈ T M is a pair in the set {(p, u) | p ∈ M, u ∈ T p M }. Let π : T M → M be the projection map. The differential of the projection map is a smooth map denoted as dπ : T T M → T M . For any vector fields X, Y ∈ X (M ), the Levi-Civita connection on M is denoted by ∇ X Y . From Sasaki, the tangent space TP T M is a direct sum decomposition TP T M = HP VP , where HP is the horizontal subspace and VP is the vertical subspace [6]. To construct the subspaces, we begin by defining the exponential map on M . For an open neighborhood U of p := π P ∈ M , the exponential map exp p : to the Levi-Civita connection is defined as The vertical subspaces is then defined as the kernel of the differential dπ, while the horizontal subspace is defined as the kernel of the connection map K. Throughout this paper, We will use the dπ and K mappings as projections on the horizontal and vertical subspaces. A curveγ : I → T M in the tangent bundle is said to be horizontal if its tangent γ ′ (t) satisfiesγ ′ (t) ∈ Hγ (t) for all t ∈ I. And similarly, a curveγ : If X is a vector field on M , then there is a unique vector field Xh on T M called the horizontal lift of X and a unique vector field Xv on T M called the vertical lift of X such that dπ Xh P = X π(P ) , K Xh P = 0 π(P ) dπ Xv P = 0 π(P ) , K Xv P = X π(P ) (2.2) for allP ∈ T M . A result of the tangent space decomposition is that any tangent vectorX ∈ TP T M can be decomposed into its horizontal and vertical components It is important to note that the standard results and our results in section 3 rely on vector fields that only change along horizontal curves. We will denote these type of vector fields as lift-decomposable vector fields. Remark 2.2. Lift-decomposable vector fields are constant along the fibers in the that sense dπ X (p, u) (p,u) = dπ X (p, u ′ ) (p,u ′ ) and similarly for the connection map K ofX for any p ∈ M, u, u ′ ∈ T p M , andX ∈ X (T M ).
In general, lift-decomposable vector fields may be too limiting. In section 4, we show how to extend the results for lift-decomposable vector fields to any general vector fields that may change along both horizontal and vertical curves.
As shown in [2], the Lie bracket of horizontal and vertical lifts on T M are given by the following for all vector fields X, Y ∈ X (M ), (p, u) ∈ T M , Xh, Xv, Yh, Yv are the respective horizontal and vertical lifts, and R is the curvature tensor on M . Note that the vector fields are lift decomposable.
A metricḡ on the tangent bundle is said to be natural with respect to g on M if for all X, Y ∈ X (M ) and (p, u) ∈ T M . The Sasaki metric, first introduced in [6], is a special natural metric that has been widely used to study the relationship between the base manifold and its tangent bundle. The Sasaki metric is given as The Kozul formula on (M, g) is given by for all vector fields X, Y, Z ∈ X (M ).

Levi-Civita Connection for Lift-Decomposable Vector Fields .
In this section, we assume that all vector fields on T M are lift decomposable. We define a non-natural metricḡ (p,u) on the tangent bundle as Proof. The metricḡ must be an inner product on T (p,u) T M at each point (p, u) ∈ T M . The symmetry and linearity properties can be verified through simple calculations. To show positive definiteness, we consider a tangent vectorZ ∈ T (p,u) T M whereZ = Xh + Yv for X, Y ∈ T p M . Then The metric can be bound from below by using the Cauchy-Schwarz inequality such thatḡ where · is the norm with respect to g. The above equation can be rewritten in matrix notation asḡ From the above inequality,ḡ must be positive definite since the middle matrix is positive definite by m 1 , m 3 > 0 and m 1 m 3 − m 2 2 > 0. The metric in (3.1) allows us to choose from a class of metrics on the tangent bundle with different horizontal and vertical subspaces along with their Levi-Civita connections.
Using the metric defined in (3.1), the Kozul formula in (2.6), and the relations from (2.3), we derive properties of the corresponding Levi-Civita connection∇ on T M for horizontal and vertical lifts (the proof closely mirrors those found in [2,4,5,6]).
Proof. The Kozul formula on the tangent bundle is used repeatedly to find the properties of the Levi-Civita connection.
(i) The statement follows from the Kozul formula in the first equation.
for all vector fieldsX ∈ X (T M ), (p, u) ∈ T M , and m 1 , m 2 , m 3 ∈ R such that m 1 , m 3 > 0 and m 1 m 3 − m 2 2 > 0. Then Proof. The claim follows directly from the definitions off andḡ. Lemma 3.3 is important in that if there is an expression forḡ with one known tangent vector, the horizontal and vertical components of the unknown tangent vector can be extracted through the metric instead of the dπ and K mappings.
Remark 3.4. The results of Lemma 3.3 can be better understood in local coordinates using matrix operations. To illustrate the point, we assume g to be the natural Euclidean inner product on M , then where I n is the n×n identity matrix,X,Ȳ ∈ T (p,u) T M , (p, u) ∈ T M , and m 1 , m 2 , m 3 ∈ R such that m 1 , m 3 > 0 and m 1 m 3 − m 2 2 > 0. Since M is positive definite, its inverse M −1 exists. Thus, the functionf can be interpreted (in matrix notation) as Whenf acts on a tangent vector in (3.4), we recover the identity matrix and the simple pairing of the horizontal and vertical components.
The following theorem combines Proposition 3.2 and Lemma 3.3 to extract the explicit form of the horizontal and vertical components of the Levi-Civita connection ∇XȲ on T M for any vector fieldsX,Ȳ ∈ X(T M ).
Proof. Proposition 3.2 provides an expression forḡ (p,u) ∇XȲ , · for any vector fieldsX,Ȳ ∈ X (T M ) at a point (p, u) ∈ T M where the second argument ofḡ can be chosen arbitrarily. Thus, we chose a purely horizontal and vertical field to extract the components of the connection. For any arbitrary vector field Z ∈ X(M ) andf defined in Lemma 3.3 The results of this section allows us to compute the Levi-Civita connection on the tangent bundle for any vector fieldsX,Ȳ ∈ X(T M ) that are lift decomposable. However, lift-decomposable vector fields do not span the space of all possible smooth vector fields. In general, vector fields on the tangent bundle may change along both horizontal and vertical curves. In the next section, we show how to extend the results for lift-decomposable vector fields to any general vector field.
4. Levi-Civita Connection for General Vector Fields. In this section, we extend the Levi-Civita connection in section 3 to general vector fields on the tangent bundle that may change along both horizontal and vertical curves. As discussed in section 2 and section 3, the Levi-Civita connection in Theorem 3.5 is only valid for lift-decomposable vector fields. In general, vector fieldsȲ ∈ X (T M ) at a point (p, u) ∈ T M depend on both horizontal and vertical motions and may be expressed as where A, B ∈ T p M ,C ∈ H (p,u) ,D ∈ V (p,u) , andC =D = 0 at (p, u) and along horizontal curves passing through (p, u). To be more precise,C andD are the pointwise horizontal and vertical projections of the fieldȲ (p ′ ,u ′ ) − Ah (p,u) − Bv (p,u) for any point (p ′ , u ′ ) ∈ T M in the neighborhood around (p, u). It is important to note that A and B change along horizontal curves, andC andD change along vertical curves.
The standard results and our results in section 3 already considered how vector fields change along horizontal curves to derive the connection in Theorem 3.5. In that formulation, we ignored the motion along the vertical curves because the vector fields are lift decomposable and thus constant along those curves. Now, we must also consider changes along vertical curves to obtain the Levi-Civita connection for general vector fields on the tangent bundle.  Proof. The proof follows from the vector field decomposition in (4.1) and the properties of the Levi-Civita connection. Note that sinceC,D = 0 along horizontal curves, the connection∇ Fh C +D = 0.

Examples.
In this section, we present two applications of our results. In the first, we show that the Sasaki metric and the corresponding Levi-Civita connection on T M is a special case. In the second, we apply the results on SO(3) and derive the Levi-Civita connection on T SO(3).

Sasaki Metric.
In this example, we show that the Saski Metric [6] and the induced Levi-Civita connection on T M is a special case of our results. If we choose m 1 = 1, m 2 = 0, m 3 = 1 then the metric (3.1) becomesḡ The induced connection∇ on T M , given by Theorem 3.5, can be shown to be equivalent to the results obtained by Kowalski in [5].

SO(3) Example.
In this example, we consider the Special Orthogonal Group SO(3) equipped with a metric g and its tangent bundle T SO(3) equipped with the metric in (3.1). The Levi-Civita connection on SO(3) is given by Edelman in [3] for all vector fields X, Y ∈ X (SO (3)) at a point R ∈ SO(3) andẎ is the usual time derivative. Given left-invariant vector fieldsX,Ȳ ∈ X (T SO (3)) along a curveγ such that where constants ζ, η, α, β, ω ∈ R 3 ,( ·) : R 3 → so (3)  In the general case where ω = ω(t), α = α(ω), β = β(ω), an additional term is required to account for changes in the vertical subspace along the curveγ (see where∇ is the usual directional derivative on R 3 . Both results can be validated by the metric compatibility requirement along their respective curves.
6. Conclusions. In this paper, we study the relationship between Riemannian manifolds and their tangent bundle. Namely, we see that a manifold equipped with a metric and Levi-Civita connection induces a metric and Levi-Civita connection on its tangent bundle by the natural decomposition of the tangent bundle into the horizontal and vertical subspaces. We then defined a non-natural metric on the tangent bundle and derived the corresponding Levi-Civita connection. In addition, we showed explicitly how to extend the results to vector fields that are not constant along the fibers. As a validation of our results, we see that under special conditions the non-natural metric reduces to the Sasaki metric and the corresponding Levi-Civita connection agrees with the results of Kowalski.