Smooth Manifolds, Tangent and Cotangent Bundles, and Tensor Fields

Charts make calculus possible locally, but no individual chart is preferred. Compatible transition maps translate between coordinate descriptions; tangent and cotangent spaces capture directional and linear response at a point; tensor fields assemble multilinear data smoothly over the manifold; and a Lie derivative compares a tensor with its pullback along a local flow. The objects are coordinate-independent even though every calculation may use coordinates.

These structures are the kinematic language for classical fields on spacetime. Field equations, global causal evolution, and quantization are not developed here.

This page works with finite-dimensional, Hausdorff, second-countable smooth manifolds, normally without boundary. No metric, connection, orientation, or volume form is assumed.

A smooth structure is local calculus that glues

An $n$-dimensional topological manifold is a Hausdorff, second-countable space in which every point has a neighborhood homeomorphic to an open subset of $\mathbb R^n$. A chart $(U,x)$ consists of such a neighborhood and a homeomorphism

$$x:U\longrightarrow x(U)\subset\mathbb R^n, \qquad p\longmapsto \bigl(x^1(p),\ldots,x^n(p)\bigr).$$

Two charts $(U,x)$ and $(V,y)$ are smoothly compatible when either they do not overlap or the transition map

$$y\circ x^{-1}: x(U\cap V)\longrightarrow y(U\cap V)$$

is a smooth diffeomorphism. A compatible atlas covering $M$ determines a unique maximal compatible atlas, called a smooth structure. The pair of the underlying topological manifold and this smooth structure is a smooth manifold.

This definition separates what is local from what is geometric:

• $x^\mu$ and $y^a$ are temporary numerical labels;
• the smooth transition maps are the rules that make derivatives computed with different labels agree; and
• the manifold point $p$ is not its coordinate tuple $x(p)$.

For example, the two-sphere is smooth even though no single chart covers it. Stereographic charts from opposite poles have a smooth transition map on their overlap. By contrast, polar variables are not a chart at the origin: the angle is not defined there, and the coordinate Jacobian degenerates. Away from the origin an angular cut is still needed to make the angle single-valued. This is a failure of a proposed coordinate system, not a singularity of the plane.

Let $F:M\to N$ be a map. It is smooth at $p$ when there are charts $(U,x)$ about $p$ and $(V,y)$ about $F(p)$ such that $F(U)\subseteq V$ and the local representative

$$y\circ F\circ x^{-1}$$

is smooth on the open set $x(U)$. The map is smooth when it is smooth at every point. Compatibility of the atlases makes this condition independent of the charts selected. This definition also implies that $F$ is continuous. A bijective smooth map whose inverse is smooth is a diffeomorphism.

A chart change is passive: it changes the numbers used to represent a point or field. A diffeomorphism can instead be used actively to move points and fields. Keeping those operations distinct prevents many apparent contradictions about how components transform. Lee gives the structural development in Lee 2013, Chapters 1–3; Nakahara gives a physics-facing treatment in Nakahara 2003, §§ 5.1–5.2.

Tangent vectors turn directions into derivations

At $p\in M$, a tangent vector can be defined without embedding $M$ in a larger space. It is a linear map

$$v:C^\infty(M)\longrightarrow\mathbb R$$

that obeys the Leibniz rule at $p$,

$$v(fg)=f(p)v(g)+g(p)v(f).$$

The vector space of these derivations is the tangent space $T_pM$. If a smooth curve $\gamma$ satisfies $\gamma(0)=p$, then

$$v_\gamma(f) = \left.\frac{d}{ds}\right|_{s=0}f\bigl(\gamma(s)\bigr)$$

is a tangent vector. Conversely, every tangent vector is the velocity of a curve in this sense. Thus curve velocities and derivations are equivalent models of $T_pM$.

On a chart domain, the coordinate derivations

$$\left.\partial_\mu\right|_p \equiv \left.\frac{\partial}{\partial x^\mu}\right|_p$$

form a basis, so $v=v^\mu_{(x)}\partial_\mu|_p$. Under $y=y(x)$,

$$\left.\frac{\partial}{\partial y^a}\right|_p = \left. \frac{\partial x^\mu}{\partial y^a} \right|_p \left.\frac{\partial}{\partial x^\mu}\right|_p, \qquad v^a_{(y)} = \left. \frac{\partial y^a}{\partial x^\mu} \right|_p v^\mu_{(x)}.$$

The basis and components change together, leaving $v$ unchanged. Coordinate bases exist only where their charts do, and the smooth structure alone gives no canonical way to identify $T_pM$ with $T_qM$ when $p\ne q$.

A smooth map $F:M\to N$ induces the differential, or pushforward at a point,

$$dF_p:T_pM\longrightarrow T_{F(p)}N, \qquad \bigl(dF_pv\bigr)(h)=v(h\circ F), \quad h\in C^\infty(N).$$

This is the coordinate-free chain rule. In coordinates its matrix is the Jacobian of the local representative of $F$.

Cotangent vectors measure tangent vectors

The cotangent space

$$T_p^*M=\operatorname{Hom}(T_pM,\mathbb R)$$

is the linear dual of $T_pM$. For $f\in C^\infty(M)$, its differential at $p$ is the covector

$$df_p(v)=v(f).$$

The basis $dx^\mu|_p$ is dual to $\partial_\mu|_p$, and a covector has the form $\alpha=\alpha^{(x)}_\mu dx^\mu|_p$. On a chart overlap,

$$dy^a = \frac{\partial y^a}{\partial x^\mu}\,dx^\mu, \qquad \alpha^{(y)}_a = \frac{\partial x^\mu}{\partial y^a}\, \alpha^{(x)}_\mu.$$

Therefore the pairing is invariant:

$$\alpha(v) = \alpha^{(x)}_\mu v^\mu_{(x)} = \alpha^{(y)}_a v^a_{(y)}.$$

The differential of a map naturally reverses direction on covectors. If $\alpha$ is a covector field on $N$, its pullback is

$$\bigl(F^*\alpha\bigr)_p(v) = \alpha_{F(p)}\bigl(dF_pv\bigr).$$

More generally, an arbitrary smooth map naturally pulls back covariant tensors. It does not naturally pull back contravariant or mixed tensors; that operation is available when $F$ is a diffeomorphism.

The disjoint unions

$$TM=\bigsqcup_{p\in M}T_pM, \qquad T^*M=\bigsqcup_{p\in M}T_p^*M$$

carry natural smooth-manifold and vector-bundle structures. A vector field is a smooth section $X:M\to TM$ with $X(p)\in T_pM$; a covector field is a smooth section of $T^*M$. Their coordinate components are smooth functions, and smoothness in one compatible atlas implies smoothness in every compatible chart.

Tensor fields are global objects, not component arrays

This page uses the convention that a tensor of type $(r,s)$ at $p$ is an element of

$$T^r{}_s(T_pM) = (T_pM)^{\otimes r}\otimes(T_p^*M)^{\otimes s}.$$

Equivalently, it is multilinear in $r$ covectors and $s$ vectors. A tensor field of type $(r,s)$ is a smooth section of

$$T^r{}_sM = (TM)^{\otimes r}\otimes(T^*M)^{\otimes s}.$$

Functions, vector fields, and covector fields have types $(0,0)$, $(1,0)$, and $(0,1)$ respectively. A metric, when one is later supplied, is a nondegenerate symmetric tensor field of type $(0,2)$.

In a chart,

$$T = T^{\mu_1\cdots\mu_r}{}_{\nu_1\cdots\nu_s} \, \partial_{\mu_1}\otimes\cdots\otimes\partial_{\mu_r} \otimes dx^{\nu_1}\otimes\cdots\otimes dx^{\nu_s}.$$

On an overlap with coordinates $y=y(x)$, the same field has components

$$\begin{aligned} T^{a_1\cdots a_r}{}_{b_1\cdots b_s} ={}& \frac{\partial y^{a_1}}{\partial x^{\mu_1}} \cdots \frac{\partial y^{a_r}}{\partial x^{\mu_r}} \\ &\times \frac{\partial x^{\nu_1}}{\partial y^{b_1}} \cdots \frac{\partial x^{\nu_s}}{\partial y^{b_s}} T^{\mu_1\cdots\mu_r}{}_{\nu_1\cdots\nu_s}. \end{aligned}$$

All Jacobians are evaluated at the same point. Every upper index contributes a forward Jacobian and every lower index an inverse Jacobian. Tensor products, permutations of slots, and contractions preserve this rule. Consequently a fully contracted scalar has the same value in every chart.

A family of arrays is therefore not automatically a tensor field. It represents one only when its arrays are smooth and obey the overlap law. Likewise, the individual functions $\partial_\rho T^{\mu_1\cdots\mu_r}{}_{\nu_1\cdots\nu_s}$ usually do not transform as components of a tensor: derivatives of the Jacobians produce extra terms under nonlinear coordinate changes. A connection supplies the correction needed for covariant differentiation, but that is additional structure.

Nor does the smooth structure canonically identify $TM$ with $T^*M$. Raising and lowering indices requires a metric. This is why an upper index and a lower index record genuinely different kinds of input here, rather than a typographical choice. Frankel 2012, Chapters 1–2 develops these transformation laws and the failure of raw component derivatives; Lee, Chapters 10–12, supplies the vector-bundle, cotangent, and tensor constructions.

Flows compare tensors without a connection

Let $X$ be a smooth vector field. Through each point there is a unique maximal integral curve, at least for a sufficiently small parameter interval. Together these curves define a local flow

$$\Phi_s(p), \qquad \Phi_0(p)=p, \qquad \frac{d}{ds}\Phi_s(p)=X_{\Phi_s(p)}$$

wherever the curve is defined. On their common domains, $\Phi_{s+t}=\Phi_s\circ\Phi_t$; for fixed $s$, the map $\Phi_s$ is a diffeomorphism between appropriate open subsets, with inverse $\Phi_{-s}$. Completeness of $X$ would make $\Phi_s$ available on all of $M$ for every $s\in\mathbb R$, but completeness is not required below.

For a tensor field $T$, the local-flow diffeomorphism $\Phi_s$ pulls the moved field back to the original point, where it can be compared with $T$. The Lie derivative is

$$\mathcal L_XT = \left. \frac{d}{ds} \right|_{s=0} \Phi_s^*T,$$

where the local flow is defined. No metric or connection is needed. The basic cases are

$$\mathcal L_X f=X(f), \qquad \mathcal L_XY=[X,Y],$$

and

$$(\mathcal L_X\alpha)_\mu = X^\rho\partial_\rho\alpha_\mu + \alpha_\rho\partial_\mu X^\rho.$$

For a general type-$(r,s)$ tensor, the coordinate formula is

$$\begin{aligned} (\mathcal L_XT)^{\mu_1\cdots\mu_r}{}_{\nu_1\cdots\nu_s} ={}& X^\rho\partial_\rho T^{\mu_1\cdots\mu_r}{}_{\nu_1\cdots\nu_s} \\ &- \sum_{i=1}^{r} (\partial_\rho X^{\mu_i}) T^{\mu_1\cdots\rho\cdots\mu_r}{}_{\nu_1\cdots\nu_s} \\ &+ \sum_{j=1}^{s} (\partial_{\nu_j}X^\rho) T^{\mu_1\cdots\mu_r}{}_{\nu_1\cdots\rho\cdots\nu_s}. \end{aligned}$$

The separate partial-derivative terms are coordinate-dependent, but their full combination is tensorial. If $\mathcal L_XT=0$ throughout the relevant domain, then $T$ is invariant under the local flow generated by $X$. This derivative is different from a covariant derivative: the Lie derivative uses the differential of $X$‘s own flow, whereas a connection supplies infinitesimal comparison between neighboring fibers along a chosen tangent direction. Nakahara, § 5.3, treats flows and Lie derivatives of general tensor fields; Frankel, Chapter 4, gives the corresponding physics-facing treatment for vector fields and forms.

Controlled example: a scalar field in two coordinate systems

Use $M=\mathbb R^2$ as the smooth manifold underlying a $1+1$-dimensional spacetime. Introduce global coordinates $(t,x)$ and the linear coordinates

$$u=t-x, \qquad v=t+x,$$

whose inverse is $t=(u+v)/2$, $x=(v-u)/2$. This calculation uses only the smooth structure; no metric or field equation is being assumed.

Consider the kinematic scalar-field configuration

$$\phi=f(u)$$

for a smooth function $f$. Its differential is the same covector field in either chart:

$$d\phi = f'(u)\,du = f'(t-x)(dt-dx).$$

Now take

$$X=\partial_v = \frac12\bigl(\partial_t+\partial_x\bigr).$$

In $(u,v)$ coordinates the components of $d\phi$ and $X$ are $(f'(u),0)$ and $(0,1)$. In $(t,x)$ coordinates they are $(f'(t-x),-f'(t-x))$ and $(1/2,1/2)$. Both arrays give the same contraction:

$$d\phi(X)=X(\phi)=0.$$

The covariant tensor

$$K=d\phi\otimes d\phi$$

has $K_{uu}=(f'(u))^2$ as its only potentially nonzero component in $(u,v)$ coordinates. In $(t,x)$ coordinates,

$$K_{tt}=K_{xx}=(f'(t-x))^2, \qquad K_{tx}=K_{xt}=-(f'(t-x))^2.$$

These different arrays represent the same tensor. The flow of $X$ is $(u,v)\mapsto(u,v+s)$, so it leaves $u$, $\phi$, and $K$ unchanged:

$$\mathcal L_X\phi=0, \qquad \mathcal L_XK=0.$$

The example establishes a kinematic fact—coordinate-independent field data and a flow symmetry—not a dynamical claim. Lorentzian causal structure, Cauchy evolution, and quantization require further hypotheses and belong to Curved Spacetimes, Cauchy Surfaces, and Global Hyperbolicity; Bär, Ginoux, and Pfäffle 2007, Chapters 3–4 give the analytic and quantization hypotheses behind that handoff.

Boundaries of the construction

The smooth structure supports chart changes, differentials, tangent and cotangent bundles, tensor algebra, and Lie derivatives of tensor fields. Several familiar operations require more:

• a metric identifies vectors with covectors and defines lengths, causal character, and index raising or lowering;
• a connection defines covariant derivatives and parallel transport;
• an orientation together with a top-degree form—or, alternatively, a density without an orientation—is needed for integration; and
• a general matter field may be a section of another vector bundle rather than an ordinary spacetime tensor.

In particular, a base-manifold flow acts canonically on natural tensor bundles. An arbitrary vector-bundle section needs a compatible lifted action before an analogous Lie derivative is defined. Fields carried by other vector bundles therefore require the later bundle constructions rather than being treated as ordinary spacetime tensor fields by default.

Common pitfalls

Treating components as the object. A component array is tied to a chart and basis. The tensor is the multilinear geometric object represented by all compatible arrays.

Confusing a coordinate failure with a geometric singularity. Polar coordinates fail at the origin even though the plane is smooth there. Inspect overlap maps or use another chart before drawing a geometric conclusion.

Equating vectors and covectors without a metric. They are dual spaces, not canonically the same space. A formula that silently raises or lowers an index has already introduced metric data.

Using componentwise partial derivatives as a tensor. Under nonlinear coordinate changes, derivatives of the Jacobian create extra terms. Use a Lie derivative when a flow is the relevant comparison, or introduce a connection when directional comparison between fibers is required.

Assuming every local flow is global. Smoothness gives local integral curves. A vector field must be complete before its flow is defined for every real parameter value.

Exercises

Use these brief calculations to check the two coordinate-change ideas above.

Compatibility check. On $\mathbb R$, let $x(p)=p$ and $z(p)=p^3$. Why does $z$ fail to be a coordinate smoothly compatible with $x$ near the origin, even though it is a homeomorphism?

Compatibility answer

The transition $z\circ x^{-1}(x)=x^3$ is smooth, but its inverse $x=z^{1/3}$ is not differentiable at $z=0$. Equivalently, the Jacobian of the forward transition vanishes there, so the transition is not a smooth diffeomorphism near the origin.

Component check. With $u=t-x$ and $v=t+x$, rewrite $V=t\partial_t+x\partial_x$ in the $(u,v)$ coordinate basis.

Component answer

Because $V(u)=u$ and $V(v)=v$,

$$V=u\partial_u+v\partial_v.$$

This also follows from $\partial_t=\partial_u+\partial_v$ and $\partial_x=-\partial_u+\partial_v$.

Synthesis and next steps

A smooth atlas makes local calculus consistent. Tangent vectors are coordinate-free derivations, cotangent vectors are their duals, and tensor fields are smooth sections whose component laws guarantee that contractions and other tensorial statements do not depend on the chart. A vector field’s local flow provides the active comparison used by the Lie derivative.

The next mathematical structure depends on the operation needed:

• Differential Forms, Integration, Orientation, and Stokes Theorem develops alternating covariant tensors, exterior differentiation, and integration;
• Metrics, Volume Forms, Hodge Star, and Laplace Operators introduces metric duality and metric-dependent operators;
• Vector, Principal, and Associated Bundles generalizes the tangent and cotangent examples to fields with nontrivial fiber data; and
• Bundle Connections, Curvature, Gauge Transformations, and Bianchi Identities supplies covariant differentiation on such bundles.

For the first physical application, continue to Curved Spacetimes, Cauchy Surfaces, and Global Hyperbolicity, where Lorentzian causality and predictive field evolution become part of the problem.

References

• Christian Bär, Nicolas Ginoux, and Frank Pfäffle, Wave Equations on Lorentzian Manifolds and Quantization, European Mathematical Society, 2007, Chapter 1, Appendix A.3, and Chapters 3–4. This supports the bridge from smooth manifolds and bundle-valued linear fields to globally hyperbolic evolution and algebraic quantization.
• Theodore Frankel, The Geometry of Physics: An Introduction, third edition, Cambridge University Press, 2012, §§ 1.2–1.4, 2.1–2.4, and 4.1–4.2. This is the physics-facing teaching source for manifolds, tangent and cotangent data, tensor transformation laws, local flows, and Lie derivatives of vector fields and forms.
• John M. Lee, Introduction to Smooth Manifolds, second edition, Graduate Texts in Mathematics 218, Springer, 2013, Chapters 1–3 and 8–12. This is the structural reference for smooth manifolds and maps, tangent vectors, vector fields and flows, bundles, and tensor fields.
• Mikio Nakahara, Geometry, Topology and Physics, second edition, Institute of Physics Publishing, 2003, §§ 5.1–5.3. This supplies an independent physics-facing treatment, including the Lie derivative of general tensor fields.