On the geometry of frame bundles

Intrinsic incremental evolution of hypoelastic discrete mechanical systems

“…This property means that, at any step σ and for any infinitesimal rigid displacement $$\delta _r m_e(\sigma )\in R(m_e(\sigma )){\mathbb {M}}$$: $$$$\begin{equation*} \langle F_{\mbox{{ext}}}(\sigma )(m_e(\sigma )),\delta _r m_e(\sigma )\rangle = \langle \mathbf {f}_{\mbox{{ext}}}(\sigma ),\delta _r m_e(\sigma )\rangle = 0 \end{equation*}$$$$We cannot directly transfer this property to infinitesimal forces $$\delta \mathbf {f}_{\mbox{{ext}}}\in T_{\mathbf {f}_{\mbox{{ext}}}}T^*{\mathbb {M}}$$ since such a duality bracket should held between $$TT{\mathbb {M}}$$ and $$T^*T{\mathbb {M}}=T^*(T{\mathbb {M}})$$ whereas $$\delta \mathbf {f}_{\mbox{{ext}}}\in TT^*{\mathbb {M}}= T(T^*{\mathbb {M}})$$. Due to Tulczyjew (see [8] or [6]), there is a canonical isomorphism …”

“…On the less usual topic of double vector bundles which is the core of the present paper, the above references [4] and [5] only mention the double tangent space $$TT{\mathbb {M}}$$ of a manifold $${\mathbb {M}}$$. A general presentation of double vector bundles is provided for example in [6] but the reader of the paper may omit the general theory of double vector bundles.…”

On the Differential Geometry of Coframe Bundle with Cheeger-Gromoll Metric

“…In [2], a new class of g-natural metrics was introduced on the cotangent bundle, to which the Cheeger-Gromoll metric belongs. A similar approach was implemented by K. Niedzialomski [11], applied to the bundle of linear frames.…”