Tangent space

The tangent space at a point of a smooth manifold is a vector space whose elements represent directions of motion through that point. It extends the idea of a tangent plane to curved spaces of any dimension.

Smooth manifold: A space that locally resembles Euclidean space and has smoothly compatible coordinate charts. Tangent spaces are assigned to points of these spaces.
Tangent bundle: The space formed by collecting the tangent spaces of all points of a manifold. It assembles the pointwise spaces into one global geometric object.
Riemannian metric: A smoothly varying inner product on the tangent spaces of a manifold. It measures lengths and angles of tangent directions.
Cotangent space: The dual vector space of a tangent space, consisting of linear functionals on tangent vectors. It pairs with tangent directions but contains covectors rather than vectors.
Coordinate chart: A homeomorphism from part of a space to an open subset of Euclidean space, used to assign local coordinates. Charts provide coordinates for computing tangent vectors.
Pushforward (differential): The linear map induced on tangent spaces by a smooth map between manifolds. It transports tangent vectors through smooth maps.
Vector field: A choice of tangent vector at every point of a manifold, varying smoothly. Vector fields are sections that assign elements of each tangent space.
Normal space: The subspace of ambient directions orthogonal to a submanifold's tangent space. It captures directions transverse to the manifold rather than directions along it.
Smooth curve: A differentiable map from an interval into a smooth manifold. The velocity of a curve through a point defines a tangent vector there.
Tangent vector: An element of a tangent space, representing an infinitesimal direction at a point. These are the vectors the space contains.

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