Smooth Manifolds, Maps, and Tensor Fields

Spacetime geometry begins before distances, light cones, or curvature exist. Its first task is to say what a smooth space is, what counts as a direction at one point, and which constructions survive a change of coordinates. This chapter develops that metric-free foundation through one recurring test: describe the sphere with overlapping charts, calculate in either chart, and verify that the result names the same geometric object. After the appropriate route, you will be able to distinguish points from coordinates, vectors from their components, and natural tensor operations from constructions that require extra geometry.

Required background. Calculus and Linear Algebra for Relativity — review multivariable derivatives, Jacobians, rank, dual spaces, and multilinear maps before working the diagnostic below.

Enter the smooth category

A manifold is locally modeled on $\mathbb R^n$, but its global shape need not fit into one coordinate system. Smooth transition maps on chart overlaps make differentiation consistent. Hausdorffness and second countability supply the global regularity assumed throughout this volume. None of these data determines a metric: at this stage there is no invariant length, angle, timelike direction, volume, or preferred comparison of vectors at separate points.

The smooth-manifold and atlas page makes this definition concrete with the north and south stereographic charts of $S^2$. Their inversion transition is smooth away from the omitted poles, and its cocycle identities illustrate how local descriptions glue. The line with two origins provides a discriminating failure case: it looks locally Euclidean but is not Hausdorff. These hypotheses and their consequences are treated systematically in Gualtieri 2017, §§1.1–2.5, pp. 2–23, PDF.

Directions, motions, and maps

The tangent and cotangent page constructs a tangent vector twice: as the velocity class of a curve and as a derivation of smooth functions. Their equivalence explains why a vector is geometric while its components depend on a chart. Cotangent vectors are linear measurements of directions; $df_p(v)=v(f)$ needs no metric. This separation matters later because identifying vectors with covectors requires a nondegenerate metric.

A smooth vector field supplies one tangent vector at every point. Its integral curves generate a local motion. On the flows and Lie brackets page, the fields

$$X=\partial_x, \qquad Y=x\partial_y$$

have noncommuting flows, and the leading endpoint difference is measured by $[X,Y]=\partial_y$. The calculation fixes the meaning and sign of the bracket without introducing a connection. It also separates local existence of a flow from completeness for all parameter values.

The smooth maps, pushforwards, and pullbacks page then asks what a map transports. Its differential pushes an individual tangent vector forward. Covariant tensors pull back along any smooth map because their arguments can first be pushed forward. A whole vector field, by contrast, generally cannot be pushed through a many-to-one map: different points in one fiber may prescribe inconsistent target vectors. This asymmetry is a structural fact, not notational preference.

Tensorial operations before a metric

The tensor fields and natural operations page organizes multilinear fields by their contravariant and covariant slots. Tensor products, permutation of like slots, weighted symmetrization and antisymmetrization, and contraction of one vector with one covector slot are canonical. The chapter uses

$$T_{(ij)}=\frac12(T_{ij}+T_{ji}), \qquad T_{[ij]}=\frac12(T_{ij}-T_{ji}).$$

There is still no canonical raising or lowering of indices, no trace of two covariant indices, and no covariant derivative. A useful tensoriality test is $C^\infty(M)$-linearity in every vector argument. It accepts differential forms and rejects the Lie bracket, whose value depends on derivatives of the fields. Tong 2019, §§2.1–2.3, pp. 49–75, PDF shows how this invariant hierarchy enters relativity calculations.

Finally, the embedded and immersed submanifolds page distinguishes local rank from global placement. A regular level $F^{-1}(q)$ is an embedded submanifold with tangent space $\ker dF$, whereas an immersion can cross itself. The unit sphere passes the regular-value test; a figure-eight immersion and the singular level $x^2-y^2=0$ expose different ways an image can fail to be embedded.

Choose a route

For a first general-relativity course, read the pages in the order shown. The atlas page supplies coordinate independence; tangent spaces make differentiation intrinsic; flows and maps clarify transport under diffeomorphisms; tensors collect the objects later acted on by metrics and connections; submanifolds prepare hypersurfaces.

For targeted repair, choose the shortest path:

• If coordinate transformations are the difficulty, read the atlas page and then tangent spaces.
• If active versus passive motion is unclear, read smooth maps before flows and compare pullback with the Lie bracket.
• If index manipulations feel formal, work the natural-operations page and identify exactly where a metric would be needed.
• If constraints define your geometry, read smooth maps followed by the regular-value theorem on the submanifold page.

The chapter intentionally excludes metric geometry, parallel transport, curvature, and general point-set topology. Those subjects require additional structure or a different scope. Its purpose is narrower and indispensable: establish which statements already make sense on a smooth manifold.

Check your preparation

On $S^2\subset\mathbb R^3$, let $F(X,Y,Z)=X^2+Y^2+Z^2$ and let $p\in S^2$. Explain why $S^2=F^{-1}(1)$ is a smooth embedded surface, determine $T_pS^2$, and state whether the calculation itself defines an intrinsic metric on the sphere.

Solution

At every point of $F^{-1}(1)$,

$$dF_p(v)=2(Xv_X+Yv_Y+Zv_Z)$$

is a nonzero linear functional and hence is surjective onto $\mathbb R$. Therefore $1$ is a regular value, so the regular-value theorem makes $S^2$ an embedded two-manifold. Its tangent space is

$$T_pS^2=\ker dF_p =\{v\in\mathbb R^3:Xv_X+Yv_Y+Zv_Z=0\}.$$

The ambient Euclidean dot product provides a convenient way to write the kernel as $p\mathbin{\cdot}v=0$, but the manifold and tangent-space conclusion follows from $dF$ and rank. An intrinsic metric on $S^2$ would be additional data, even though the ambient metric can later be pulled back to supply one.

What this chapter enables

You should now be able to translate between invariant definitions and coordinate expressions, compute tangent and cotangent transformations, derive a Lie bracket from flows, decide whether a pullback or pushforward exists, test an expression for tensoriality, and apply the regular-value theorem with its rank hypothesis visible. The next chapter adds differential forms, exterior differentiation, orientation, and integration while preserving this metric-free discipline.

References

• Gualtieri, Marco. Geometry and Topology I. University of Toronto course notes, 2017, §§1.1–2.5, pp. 2–23; §4.2, pp. 45–47, PDF.
• Tong, David. General Relativity. University of Cambridge lecture notes, 2019, §§2.1–2.3, pp. 49–75, PDF.