---- > [!definition] Definition. ([[canonical lift to the tangent bundle]]) > Let $M$ be a [[smooth manifold]] and $TM$ its [[tangent bundle]]. Any [[parameterized curve|curve]] $\gamma (t):I \to M$, written in [[coordinate chart|coordinates]] as $\big( x^{i}(t) \big)_{i=1}^{\text{dim }M}$, has a canonical [[lifting|lift]] to $TM$ given in coordinates as $\big( x^{i}(t), a^{j}(t) \big)_{i,j=1}^{\text{dim } M}$ where $a^{j}(t)=\dot{x}^{j}(t)$. > > > ![[Pasted image 20250518105533.png|500]] > > Cartoon: canonical lift of two curves on $\mathbb{S}^{2}$ to the four-dimensional [[tangent bundle]] $T\mathbb{S}^{2}$, visualized as two-dimensional arrows attached to two-dimensional points. ---- #### ---- #### References > [!backlink] > ```dataview > TABLE rows.file.link as "Further Reading" > FROM [[]] > FLATTEN file.tags as Tag > WHERE Tag = "#definition" OR Tag = "#theorem" OR Tag = "#MOC" OR Tag = "#proposition" OR Tag = "#axiom" > GROUP BY Tag > ``` > [!frontlink] > ```dataview > TABLE rows.file.link as "Further Reading" > FROM outgoing([[]]) > FLATTEN file.tags as Tag > WHERE Tag = "#definition" OR Tag = "#theorem" OR Tag = "#MOC" OR Tag = "#proposition" OR Tag = "#axiom" > GROUP BY Tag > ```