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Local coordinates $x^{\mu }$ induce the coordinate-basis tangent vectors ${\partial }_{\mu }=\partial /\partial x^{\mu }$. Under a coordinate change they transform by the chain rule.

The cotangent space at $p$ is the dual vector space $T_p^{\ast }M=(T_{p}M{)}^{\ast }$. Its elements are covectors on the tangent space. In a manifold chart $x$, the basis $d{x}^i{|}_p$ is dual to ${\partial }_i{|}_p$. Under a coordinate change to $y$, the coefficients of $\alpha =a_{i}d{x}^i=b_{j}d{y}^j$ obey $b_j={\sum }_i{a}_i\partial x^i/\partial y^j$. The disjoint union of these spaces is the cotangent bundle.

At $p$ on a smooth manifold, a derivation at that point is a real-linear map $v:C_p^{\infty }(M)\to \mathbb{R}$ on germs of smooth functions with $v(fh)=f(p)v(h)+h(p)v(f)$. It defines a tangent vector. The local factorization $f-f(p)={\sum }_i(x^i-x^i(p))f_i$ gives $v(f)={\sum }_{i}v(x^i){\partial }_{i}f(p)$, so the coordinate derivations form a basis of the tangent space.

A topological manifold has dimension $n$ when its manifold charts have coordinate domains open in ${\mathbb{R}}^n$. For a smooth manifold, each tangent space has dimension $n$, and a tangent bundle has vector bundle rank $n$.

The first fundamental form of a surface $S\subseteq {\mathbb{R}}^3$ is the restriction of the ambient inner product to each tangent space $T_{p}S$. For a parametrized surface $X(u,v)$, it is

An isotropic submanifold has every tangent space an isotropic subspace of a symplectic vector space. Its dimension is at most half the ambient dimension, and equality makes it a Lagrangian submanifold.

A nonempty fixed-point set of an anti-symplectic involution is a Lagrangian submanifold. Local linearization identifies its tangent space with the $+1$ eigenspace of the derivative.

In a local exponential chart, the mixed second differential of the group commutator defines a bilinear map on the tangent space at the identity. Inversion after swapping the two group elements proves antisymmetry. Differentiating conjugation gives $[X,Y]=\mathrm{ad}(X)Y$; applying naturality of the Lie bracket to the Adjoint representation of a Lie group then proves the Jacobi identity. For a matrix group this recovers $XY-YX$ by expanding through the mixed term.

A local isometry between Riemannian manifolds is a local diffeomorphism $f$ with $f^{\ast }{g}_N=g_M$. Equivalently, its differentials are isometric linear isomorphisms between tangent spaces. It preserves the Levi-Civita connection, intrinsic sectional curvature, and affinely parametrized geodesics. A map whose differentials are merely isometric injections is an isometric immersion; the local diffeomorphism requirement distinguishes these notions.

A smooth manifold carrying a nondegenerate bilinear form on each tangent space, with one negative and all remaining positive eigenvalues, or the overall reversed-sign convention. Its nonzero null vectors have vanishing metric quadratic form.