Tensors and manifolds

A smooth n-dimensional manifold is a space covered by coordinate charts into open subsets of R^n, with smooth invertible transition maps where charts overlap, together with the usual Hausdorff and countability conditions. Each chart gives a local description. A sphere cannot be covered by one nonsingular latitude–longitude chart, but this coordinate failure does not make the poles singular points of the sphere. Coordinates may break down while the geometric object remains smooth.

A tangent vector at a point describes a directional derivative of smooth scalar functions. In a chart it is expanded in the basis of coordinate derivatives. The tangent spaces at different points are separate vector spaces; comparing their elements requires additional structure. An embedding in a larger Euclidean space can provide a useful picture, but a manifold need not be defined through an embedding. Its intrinsic dimension and smooth structure are not the same as the dimension of a chosen surrounding picture.

A covector is a linear functional on tangent vectors. The differential df of a scalar function is a covector: it takes a direction and returns the directional change of f. Under a coordinate change, its components use the inverse Jacobian, so contracting it with a vector yields the same number in either chart. A general tensor combines vector and covector slots with prescribed multilinear behavior. This is why upper and lower indices have different transformation rules.

A metric assigns a smoothly varying nondegenerate symmetric bilinear form to each tangent space. A Riemannian metric is positive definite; a Lorentzian metric has a different signature appropriate to spacetime. The metric turns component changes into invariant squared lengths and identifies vectors with covectors by lowering an index. In Cartesian Euclidean coordinates these lists happen to look identical, which can hide the conceptual distinction. In polar coordinates the distinction becomes visible immediately.

When a vector field changes from point to point, differentiating its components alone ignores the changing basis. A connection supplies a rule for differentiation and parallel transport. In coordinates its coefficients appear beside the ordinary partial derivatives. Those coefficients do not transform as tensor components by themselves; their extra transformation terms compensate for derivatives of the basis. The resulting covariant derivative does transform tensorially. For a metric, the Levi-Civita connection is singled out by metric compatibility and zero torsion.

Curvature describes the failure of parallel transport around an infinitesimal loop to return a vector unchanged. A nonzero coordinate connection does not alone establish curvature: polar coordinates on the flat plane already have nonzero connection coefficients. Conversely, no coordinate trick can remove genuine curvature throughout a region. This distinction is useful in field theory and geometric models: coordinate covariance says descriptions agree, while an actual symmetry or an isometry is an additional statement about the physical or geometric structure.