Moreover, we prove that $\mathtt{T}_{1}^{\ast}\mathtt{N}$ carries a contact metric structure and study some of its proprieties.
📌 3 Key Takeaways
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Objective & Hypothesis:Let $(\mathtt{N},\mathfrak{g})$ be a Riemannian manifold, by using the musical isomorphisms ♪ and $\natural$ induced by $\mathfrak{g}$, we built a bridge between the geometry of the tangent bundle $\mathtt{TN}$ (resp. the unit tangent sphere bundle $\mathtt{T}_{1}\mathtt{N}$) equipped with the Sasaki metric $\mathfrak{g}_{S}$ (resp. the induced Sasaki metric $\bar{\mathfrak{g}}_{S}$) and that of the cotangent bundle $\mathtt{T}^{\ast}\mathtt{N}$ (resp. the unit cotangent sphere bundle $\mathtt{T}_{1}^{\ast}\mathtt{N}$) endowed with the Sasaki metric $\mathfrak{g}_{\widetilde{S}}$ (resp. the induced Sasaki metric $\tilde{\mathfrak{g}}_{\widetilde{S}}$).
📊
Key Findings & Data:Let $(\mathtt{N},\mathfrak{g})$ be a Riemannian manifold, by using the musical isomorphisms ♪ and $\natural$ induced by $\mathfrak{g}$, we built a bridge between the geometry of the tangent bundle $\mathtt{TN}$ (resp. the unit tangent sphere bundle $\mathtt{T}_{1}\mathtt{N}$) equipped with the Sasaki metric $\mathfrak{g}_{S}$ (resp. the induced Sasaki metric $\bar{\mathfrak{g}}_{S}$) and that of the cotangent bundle $\mathtt{T}^{\ast}\mathtt{N}$ (resp. the unit cotangent sphere bundle $\mathtt{T}_{1}^{\ast}\mathtt{N}$) endowed with the Sasaki metric $\mathfrak{g}_{\widetilde{S}}$ (resp. the induced Sasaki metric $\tilde{\mathfrak{g}}_{\widetilde{S}}$).
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Conclusion & Impact:Moreover, we prove that $\mathtt{T}_{1}^{\ast}\mathtt{N}$ carries a contact metric structure and study some of its proprieties.
Let $(\mathtt{N},\mathfrak{g})$ be a Riemannian manifold, by using the musical isomorphisms ♪ and $\natural$ induced by $\mathfrak{g}$, we built a bridge between the geometry of the tangent bundle $\mathtt{TN}$ (resp. the unit tangent sphere bundle $\mathtt{T}_{1}\mathtt{N}$) equipped with the Sasaki metric $\mathfrak{g}_{S}$ (resp. the induced Sasaki metric $\bar{\mathfrak{g}}_{S}$) and that of the cotangent bundle $\mathtt{T}^{\ast}\mathtt{N}$ (resp. the unit cotangent sphere bundle $\mathtt{T}_{1}^{\ast}\mathtt{N}$) endowed with the Sasaki metric $\mathfrak{g}_{\widetilde{S}}$ (resp. the induced Sasaki metric $\tilde{\mathfrak{g}}_{\widetilde{S}}$). Moreover, we prove that $\mathtt{T}_{1}^{\ast}\mathtt{N}$ carries a contact metric structure and study some of its proprieties.
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