Euclidean manifold
The setting for tensor-analysis: ordinary N-dimensional space, treated as a set of points rather than a vector space, so that curvilinear coordinates and fields on domains have somewhere to live. The corpus holds one account, bowen-wang-vectors-tensors-vol-2, whose chapter 9 opens with it.
The definition
A Euclidean point space is a set E together with an inner product space V (the translation
space) and a map f : E × E → V such that (a) f(x,y) = f(x,z) + f(z,y) and (b) for each point x
and vector v there is exactly one y with f(x,y) = v. The dimension of E is the dimension of
V. Write f(x,y) as x − y and the two conditions become x − y = (x − z) + (z − y) and x − y = v.
Drop the inner product and the same structure is an affine space bowen-wang-vectors-tensors-vol-2.
A vector space is made a Euclidean point space by letting f(v₁, v₂) = v₁ − v₂,
but the converse needs a chosen origin. Points and displacements are different kinds of thing, which
is why a field "on" E can take vector values at each point without being a vector.
Charts
A chart at x is a pair (x̂, U) with U an open set around x and x̂ : U → Rᴺ a smooth
diffeomorphism. Its N component functions are the coordinate functions; two overlapping charts are
related by a coordinate transformation, smooth in both directions. The book stays with this "special
case" of a manifold and omits general differentiable manifolds bowen-wang-vectors-tensors-vol-2.
That choice keeps the metric and Euclidean parallelism available everywhere, which the later chapters
use to define the Levi-Civita parallelism on a hypersurface.
Hypersurfaces
A hypersurface of dimension N−1 is locally the zero set of a smooth function with nonvanishing
gradient, and n = grad f / |grad f| is its unit normal. The book builds geodesics, the exponential
map, the second fundamental form and the Gauss and Codazzi equations on this footing, so a
"surface" always sits inside a flat ambient space
bowen-wang-vectors-tensors-vol-2. Intrinsic curvature is then the failure of the surface's own
parallelism to match Euclidean parallelism.
Related
tensor-analysis · differential-forms · topology (charts are the same idea without the metric) · vector-calculus · linear-algebra · mathematics