📂Geometry

Differential Geometry of Curved Manifolds

Definition¹

Let’s denote $M$ as a Riemannian manifold, and $\frak{X}(M)$ as the set of all vector fields on $M$.

$$\frak{X}(M) = \text{the set of all vector fileds of calss } C^{\infty} \text{ on } M$$

The curvature $R$ of $M$ is a function that maps $X, Y \in \frak{X}(M)$ to $R(X, Y) : \frak{X}(M) \to \frak{X}(M)$. In this context, $R(X, Y)$ is given as follows.

$$\begin{equation} R(X, Y) Z = \nabla_{Y} \nabla_{X} Z - \nabla_{X} \nabla_{Y} Z + \nabla_{[X,Y]}Z, \quad Z \in \frak{X}(M) \end{equation}$$

Such $R$ is called the Riemannian curvature or Riemannian curvature tensor.

$\nabla$ is the Levi-Civita connection on $M$, $[ \cdot, \cdot]$ is the Lie bracket.

Explanation

In other words, $R$ maps two vector fields $X, Y$ to the function $R(X, Y)$, and then $R(X, Y)$ again maps vector field $Z$ as described by $(1)$. Therefore, it’s indeed appropriate to write it as follows.

$$R : \frak{X}(M) \times \frak{X}(M) \times \frak{X}(M) \to \frak{X}(M) \\[1em] R(X,Y,Z) = \nabla_{Y} \nabla_{X} Z - \nabla_{X} \nabla_{Y} Z + \nabla_{[X,Y]}Z, \quad Z \in \frak{X}(M)$$

However, as one can see from the value of $R(X,Y,Z)$, it neatly bundles into $Z$. Moreover, since $X, Y$ is used as the differentiating variable and $Z$ as the variable being differentiated, this notation conventionally distinguishes their roles as displayed in $R(X, Y) Z$.

Also, there is a difference in the sign conventions between textbooks for definition $(1)$, but essentially they are the same.

Repeated Notation

For the Riemann curvature tensor $R$, $R: \frak{X}(M) \times \frak{X}(M) \times \frak{X}(M) \times \frak{X}(M) \to \mathcal{D}(M)$ is defined as follows.

$$R(X, Y, Z, W) := g(R(X, Y)Z, W) = \left\langle R(X, Y)Z, W \right\rangle$$

This is called the Riemann curvature tensor. The Riemann curvature tensor introduced in the definition is $R$, and this one is also $R$. The reason for this duplication is because they are essentially the same, with only a difference of multiplying a metric.

Coordinate Representation

Let’s denote the basis of $T_{p}M$ as $\left\{ X_{i} \right\}$. $R(X_{i},X_{j})X_{k}$ is notated as follows.

$$R(X_{i},X_{j})X_{k} = \sum_{s}R_{ijk}^{s}X_{s}$$

$R(X_{i}, X_{j}, X_{k}, X_{l})$ is notated as follows.

$$R_{ijkl} = R(X_{i}, X_{j}, X_{k}, X_{l}) = g\left( R(X_{i}, X_{j})X_{k}, X_{l} \right) = \sum_{s}R_{ijk}^{s}g_{sl}$$

Example: Euclidean Space

Let’s denote as $M = \mathbb{R}^{n}$. The Euclidean space is a flat space without curvature. Therefore, we expect to obtain $R(X,Y)Z = 0$. Conversely, if this result is not obtained, we can say that definition $(1)$ does not hold any meaningful value. Let $X, Z$ be set as follows.

$$X = (X^{1}, \dots, X^{n}) = \sum X^{i}\dfrac{\partial }{\partial x_{i}} \text{ and } Z = (Z^{1}, \dots, Z^{n}) = \sum Z^{k}\dfrac{\partial }{\partial x_{k}}$$

$\nabla_{X}Z$ is as follows.

$$\nabla_{X}Z = \sum_{i,k} \left( X^{i}\dfrac{\partial Z^{k}}{\partial x_{i}} + \sum_{j}X^{i}Z^{j}\Gamma_{ij}^{k} \right) \dfrac{\partial }{\partial x_{k}}$$

In the case of Euclidean space, since $\Gamma_{ij}^{k} = 0$, we obtain the following.

$$\begin{align*} \nabla_{X} Z &= \sum_{i,k} X^{i} \left( \dfrac{\partial Z^{k}}{\partial x_{i}} \right) \dfrac{\partial }{\partial x_{k}} \\ &= \sum_{k} \left( \sum_{i} X^{i} \dfrac{\partial Z^{k}}{\partial x_{i}} \right) \dfrac{\partial }{\partial x_{k}} \\ &= \sum_{k} X Z^{k} \dfrac{\partial }{\partial x_{k}} \\ &= \left( XZ^{1}, \dots, XZ^{n} \right) \end{align*}$$

Similarly, we obtain the following.

$$\nabla_{Y} \nabla_{X} Z = \left( YXZ^{1}, \dots, YXZ^{n} \right)$$

Therefore,

$$\begin{align*} R(X, Y) Z &= \nabla_{Y} \nabla_{X} Z - \nabla_{X} \nabla_{Y} Z + \nabla_{[X,Y]}Z \\ &= \left( YXZ^{1}, \dots, YXZ^{n} \right) - \left( XYZ^{1}, \dots, XYZ^{n} \right) \\ &\quad + \left( (XY-YX)Z^{1}, \dots, (XY-YX)Z^{n} \right) \\ &= 0 \end{align*}$$

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Properties

(a) $R$ is bilinear. That is,

$$\begin{align*} R(f X_{1} + gX_{2}, Y_{1}) &= fR(X_{1}, Y_{1}) + gR(X_{2}, Y_{1}) \\[1em] R(X_{1}, fY_{1} + gY_{2}) &= fR(X_{1}, Y_{1}) + gR(X_{1}, Y_{2}) \end{align*}$$

where $f, g \in \mathcal{D}(M)$, $\quad X_{1}, X_{2}, Y_{1}, Y_{2} \in \frak{X}(M)$ are given.

(b) For any $X, Y \in \frak{X}(M)$, $R(X, Y)$ is linear. That is,

$$\begin{align*} R(X, Y) (Z + W) &= R(X, Y) Z + R(X, Y) W \\[1em] R(X, Y) fZ &= f R(X, Y) Z \end{align*}$$

where $f \in \mathcal{D}(M)$, $\quad Z, W \in \frak{X}(M)$ are given.

Proof

(b)

The first property is trivial by the definition of connection. Thus, only the second line is proven.

$$R(X, Y) fZ = \nabla_{Y} \nabla_{X} fZ - \nabla_{X} \nabla_{Y} fZ + \nabla_{[X,Y]}fZ$$

Firstly, calculating the first term by the definition of the connection,

$$\begin{align*} \nabla_{Y} \nabla_{X} (fZ) &= \nabla_{Y}(f\nabla_{X} Z + (Xf)Z) \\ &= \nabla_{Y}(f\nabla_{X}Z) + \nabla_{Y}((Xf)Z) \\ &= f\nabla_{Y}\nabla_{X}Z + (Yf)\nabla_{X}Z + (Xf)\nabla_{Y}(Z) + (Y(Xf))Z \end{align*}$$

Similarly calculating,

$$\nabla_{X}\nabla_{Y}(fZ) = f\nabla_{X}\nabla_{Y}Z + (Xf)\nabla_{Y}Z + (Yf)\nabla_{X}(Z) + (X(Yf))Z$$

Therefore, we obtain the following.

$$\begin{align*} \nabla_{Y} \nabla_{X} (fZ) - \nabla_{X}\nabla_{Y}(fZ) &= {\color{blue}f\nabla_{Y}\nabla_{X}Z} + {\color{red}\cancel{\color{black}(Yf)\nabla_{X}Z}} + {\color{green}\cancel{\color{black}(Xf)\nabla_{Y}Z}} + YXfZ \\ &\quad - \left( {\color{blue}f\nabla_{X}\nabla_{Y}Z} + {\color{green}\cancel{\color{black}(Xf)\nabla_{Y}Z}} + {\color{red}\cancel{\color{black}(Yf)\nabla_{X}Z}} + XYfZ \right) \\ &= {\color{blue}f(\nabla_{Y}\nabla_{X}Z - \nabla_{X}\nabla_{Y}Z)} + YXfZ - XYfZ \\ &= f(\nabla_{Y}\nabla_{X}Z - \nabla_{X}\nabla_{Y}Z) - ([X,Y]f)Z \end{align*}$$

Additionally, since $\nabla_{[X,Y]} fZ = f\nabla_{[X,Y]}Z + ([X,Y]f)Z$, finally we calculate as follows.

$$\begin{align*} R(X, Y) fZ &= \nabla_{Y} \nabla_{X} fZ - \nabla_{X} \nabla_{Y} fZ + \nabla_{[X,Y]} fZ \\ &= f(\nabla_{Y}\nabla_{X}Z - \nabla_{X}\nabla_{Y}Z) - ([X,Y]f)Z + f\nabla_{[X,Y]}Z + ([X,Y]f)Z \\ &= f(\nabla_{Y}\nabla_{X}Z - \nabla_{X}\nabla_{Y}Z) + f\nabla_{[X,Y]}Z \\ &= f(\nabla_{Y}\nabla_{X}Z - \nabla_{X}\nabla_{Y}Z + \nabla_{[X,Y]}Z) \\ &= fR(X, Y) Z \end{align*}$$

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