Tangent Space to a Manifold
Definition
Definition. Let be a differentiable curve on a smooth manifold . Consider the set set of all functions that are differentiable at . The tangent vector to the curve at is the operator
\dot{c}(0):&\mathcal{C}^\infty&\to&\mathbb{R}\\ &f&\mapsto&\dot{c}(0)(f)=\dfrac{d(f\circ c)}{dt}(0)\;. \end{array}$$ The _tangent space_ at $p$ is the space $T_p(M)$ of all tangent vectors at $p$. ## Remarks **Remarks**. The tagent vector satisfies, for every $\alpha,\beta\in\mathbb{R}$, and for every $f,g\in\mathcal{C}^\infty(p)$ the two conditions 1. Linearity: $\dot{c}(0)(\alpha f+\beta g)=\alpha \dot{c}(0)(f)+\beta \dot{c}(0)(g)$. 2. [[Chain rule]]: $\dot{c}(0)(fg)=(\dot{c}(0)(f))g(0)+f(0)\dot{c}(0)(g)$ The tangent space $T_pM$ is a [[vector space]] with operations in defined, for every [[differentiable]] curves $c_1,c_2\in M$ by $$(\dot{c}_1(0)+\dot{c}_2(0))f=\dot{c}_1(0)(f)+\dot{c}_2(0)(f)$$ $$\alpha (\dot{c}(0)(f))=(\alpha \dot{c}(0))(f)\;.$$ ## References - [[L. Godinho, J. Natario - An introduction to riemannian geometry|@GodNat14, Definition 4.1]] - [[Boothby, W. M. - An introduction to differentiable manifolds and riemannian geometry|@Boo75 Chapter 4 Definition 1.1]]