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Tangent Space and Tensors

"The tangent space is the right derivative of a manifold at a point – and everything geometric we measure lives on it."

On a smooth manifold (see Manifolds – an Intuitive View) functions can be differentiated. For that one needs the notion of a velocity at a point. This idea is formalised by the tangent space. On its tensor products live the concepts of length (Riemannian metric), curvature, and volume.

1. Tangent vectors

Let \(M\) be a smooth manifold, \(p \in M\). A smooth curve through \(p\) is a smooth map \(\gamma : (-\varepsilon, \varepsilon) \to M\) with \(\gamma(0) = p\). Two curves \(\gamma_1, \gamma_2\) are equivalent if in some (equivalently, every) chart $$ \frac{\mathrm{d}}{\mathrm{d}t}\Big|{t=0}(\varphi \circ \gamma_1) = \frac{\mathrm{d}}{\mathrm{d}t}\Big|(\varphi \circ \gamma_2). $$ An equivalence class is called a tangent vector at \(p\). The set of all tangent vectors is the tangent space \(T_p M\).

\(T_p M\) is a real vector space of dimension \(n = \dim M\). In a chart \((x^1, \dots, x^n)\) around \(p\) a basis is $$ \Big{ \partial_1 \big|_p, \dots, \partial_n \big|_p \Big}, \qquad \partial_i \big|_p f := \frac{\partial(f \circ \varphi^{-1})}{\partial x^i}(\varphi(p)). $$

Picture: \(T_p M\) is the tangent line/plane to \(M\) at \(p\), detached from the manifold and made into its own vector space.

2. Vector fields and the cotangent space

A vector field \(X\) assigns to every point \(p\) a vector \(X_p \in T_p M\) (smoothly). In coordinates: \(X = X^i(x)\,\partial_i\) (Einstein summation).

The cotangent space \(T_p^* M\) is the dual of \(T_p M\). A 1-form assigns to every point a linear form on \(T_p M\). The basis dual to \(\partial_i\) is the differentials \(\mathrm{d}x^i\). Then a 1-form is \(\omega = \omega_i(x)\,\mathrm{d}x^i\), and \(\mathrm{d}x^i(\partial_j) = \delta^i_j\).

3. Tensors and tensor fields

An \((r,s)\)-tensor at \(p\) is a multilinear map $$ T : \underbrace{T_p^ M \times \dots \times T_p^ M}{r} \times \underbrace{T_p M \times \dots \times T_p M} \to \mathbb{R}. $$ A tensor field assigns to every point such a tensor. In coordinates: $$ T = T^{i_1 \dots i_r}{}{j_1 \dots j_s}\, \partial \otimes \mathrm{d}x^{j_1} \otimes \dots \otimes \mathrm{d}x^{j_s}. $$} \otimes \dots \otimes \partial_{i_r

Tensor Type Examples
function \((0,0)\) scalar field
vector field \((1,0)\) \(\partial_i\)
1-form \((0,1)\) \(\mathrm{d}f\), \(\mathrm{d}x^i\)
Riemannian metric \((0,2)\), symmetric \(g_{ij}\)
curvature tensor \((1,3)\) \(R^l{}_{ijk}\)
Ricci tensor \((0,2)\), symmetric \(R_{ij}\)

4. The Riemannian metric

A Riemannian metric \(g\) is a symmetric, positive definite \((0,2)\)-tensor field. At every point \(g_p\) is an inner product on \(T_p M\), in coordinates \(g_{ij}(x)\) with \(g_{ij} = g_{ji}\) and \(g_{ij}\xi^i \xi^j > 0\) for \(\xi \neq 0\).

This yields:

  • Length of a vector: \(|v|_g = \sqrt{g_{ij} v^i v^j}\).
  • Length of a curve: \(L(\gamma) = \int_a^b |\dot\gamma(t)|_g\, \mathrm{d}t\).
  • Volume form: \(\mathrm{d}V_g = \sqrt{\det g_{ij}}\, \mathrm{d}x^1 \wedge \dots \wedge \mathrm{d}x^n\).
  • Inverse metric: \(g^{ij}\), defined by \(g^{ij}g_{jk} = \delta^i_k\).

Example: on \(\mathbb{R}^n\) the Euclidean metric is \(g_{ij} = \delta_{ij}\). On the sphere \(S^2 \subset \mathbb{R}^3\) the embedding gives, in spherical coordinates, \(g = \mathrm{d}\theta^2 + \sin^2\theta\,\mathrm{d}\phi^2\).

5. Raising and lowering indices

With \(g_{ij}\) and \(g^{ij}\) one can convert indices. From a vector field \(X^i\) one obtains the 1-form \(X_i := g_{ij} X^j\). From the curvature tensor \(R^l{}_{ijk}\) one obtains the four-index Riemann tensor \(R_{lijk} := g_{lm} R^m{}_{ijk}\). Once you get used to this mechanism you read almost every formula of differential geometry as a balance of upper/lower indices.

6. What comes next

Once a metric is chosen there is a unique torsion-free, metric compatible connection – the Levi-Civita connection – and hence covariant derivatives, geodesics and the Riemann curvature tensor. From these arise Ricci and scalar curvature, which play the central role in the Ricci flow \(\partial_t g_{ij} = -2 R_{ij}\).

Cross-references

Sources

  • Lee, John M. (2018). Introduction to Riemannian Manifolds. Springer GTM 176, 2nd ed. Ch. 1–3.
  • do Carmo, Manfredo P. (1992). Riemannian Geometry. Birkhäuser.
  • Petersen, Peter (2016). Riemannian Geometry. Springer GTM 171, 3rd ed.