Unit 5: Covariant Differentiation & Ricci's Theorem

1. The Failure of Ordinary Differentiation

In flat Euclidean space with Cartesian coordinates, the partial derivative $\frac{\partial A^i}{\partial x^j}$ of a contravariant vector transforms as a $(1, 1)$ tensor. However, under a general curvilinear coordinate change $x \to \bar{x}$:

Differentiating with respect to $\bar{x}^\beta$:

The presence of the second partial derivative $\frac{\partial^2 \bar{x}^\alpha}{\partial x^i \partial x^j}$ prevents ordinary partial derivatives from transforming as tensors.

2. Covariant Derivative of a Contravariant Vector

Definition 5.1 (Covariant Derivative of $A^i$): The covariant derivative of a contravariant vector field $A^i$ with respect to $x^j$, denoted by $\nabla_j A^i$, $A^i_{; j}$, or $A^i_{| j}$, is defined by:

Proof of Tensorial Character:

Under $x^i \to \bar{x}^\alpha$, recall that the Christoffel symbol transforms as:

When computing $\bar{\nabla}_\beta \bar{A}^\alpha = \partial_\beta \bar{A}^\alpha + \bar{\Gamma}^\alpha_{\beta\gamma} \bar{A}^\gamma$, the second-derivative terms cancel identically with those from $\partial_\beta \bar{A}^\alpha$:

Thus, $\nabla_j A^i$ is a genuine tensor of type $(1, 1)$ (once contravariant, once covariant).

3. Covariant Derivative of a Covariant Vector (1-Form)

Let $A_i$ be a covariant vector and $B^i$ an arbitrary contravariant vector. Their contraction is an invariant scalar:

For a scalar field, the directional derivative along coordinates is just the ordinary partial derivative: $\nabla_j \phi = \partial_j \phi$. By imposing the Leibniz product rule:

Expanding the right side:

Substitute $\nabla_j B^i = \partial_j B^i + \Gamma^i_{jk} B^k$:

Relabeling dummy indices in the connection term ($i \leftrightarrow k$): $A_i \Gamma^i_{jk} B^k = A_k \Gamma^k_{ji} B^i$. Therefore:

Since $B^i$ is completely arbitrary, the bracketed expression must vanish identically:

Definition 5.2 (Covariant Derivative of $A_i$):

Note the crucial minus sign preceding the Christoffel symbol for covariant indices!

1. General Formula for Tensor Fields

The derivation extending to general tensors follows systematically from the tensor product and contraction properties.

Theorem 5.1 (Covariant Derivative of Type $(r, s)$ Tensor): For an arbitrary tensor field $T^{i_1 i_2 \dots i_r}_{j_1 j_2 \dots j_s}$ of contravariant rank $r$ and covariant rank $s$, its covariant derivative with respect to $x^k$ adds a $+ \Gamma$ connection term for each contravariant index and a $- \Gamma$ connection term for each covariant index:

Examples:

1. Rank 2 Contravariant Tensor $T^{ij}$:

2. Rank 2 Covariant Tensor $T_{ij}$:

3. Mixed Tensor $T^i_j$:

2. Properties of the Covariant Derivative Operator $\nabla$

1. Linearity: For constant scalars $a, b$:

2. Leibniz Product Rule:

3. Commutation with Contraction:

The covariant derivative commutes with any contraction of indices:

For example, contracting $T^i_i$:

since the two connection terms cancel identically ($m \leftrightarrow i$).

1. Formulation of Ricci's Theorem

One of the most profound and indispensable theorems in all of Riemannian geometry is Ricci's Theorem (often called the Lemma of Ricci or Principle of Metric Compatibility).

Theorem 5.2 (Ricci's Theorem / Covariant Constancy of the Metric): In any Riemannian or Pseudo-Riemannian space with Levi-Civita connection $\Gamma^k_{ij}$, the covariant derivatives of the fundamental metric tensor $g_{ij}$, its conjugate $g^{ij}$, and the Kronecker delta $\delta^i_j$ vanish identically everywhere:

2. Line-by-Line Mathematical Proof

Part A: Proof for $g_{ij}$

Applying the covariant derivative formula for a rank $(0, 2)$ tensor:

Recall from Section 4.1 that $g_{mj} \Gamma^m_{ki} = [ki, j]$ and $g_{im} \Gamma^m_{kj} = [kj, i]$. Substituting these relations:

Now expand $[ki, j]$ and $[kj, i]$ using the definition of Christoffel symbols of the first kind:

Summing these two expressions:

Therefore:

Part B: Proof for the Kronecker Delta $\delta^i_j$

Part C: Proof for Conjugate Metric $g^{ij}$

Using the identity $g_{im} g^{mj} = \delta_i^j$ and applying the Leibniz product rule:

Since $\nabla_k g_{im} = 0$:

Multiplying by $g^{il}$:

3. Profound Consequence: Index Raising/Lowering Commutes with $\nabla$

Because $g_{ij}$ and $g^{ij}$ behave as constants with respect to covariant differentiation:

The operations of raising and lowering indices commute unconditionally with covariant differentiation!

1. Invariant Gradient of a Scalar Field

Let $\Phi(x)$ be a scalar invariant field. The covariant gradient is simply the partial derivative:

The contravariant gradient vector is obtained by raising the index:

2. Invariant Divergence

Let $V^i$ be a contravariant vector field. Its divergence is the contraction:

Recalling $\Gamma^i_{ik} = \frac{1}{\sqrt{|g|}} \partial_k \sqrt{|g|}$ from Section 4.4:

3. Invariant Curl of a Covariant Vector

Let $A_i$ be a covariant vector (1-form). Its covariant curl is defined as:

Expanding via Christoffel symbols:

Since the Levi-Civita connection is symmetric ($\Gamma^k_{ji} = \Gamma^k_{ij}$), the connection terms cancel out completely:

This demonstrates that the exterior derivative (curl) of a differential 1-form is completely independent of the connection and metric!

4. The Laplace-Beltrami Operator $\Delta \Phi$

Definition 5.3 (Laplace-Beltrami Operator): The Laplace-Beltrami operator $\Delta \Phi$ (or $\nabla^2 \Phi$) acting on a scalar field $\Phi$ is the invariant divergence of its contravariant gradient:

In coordinate form:

Orthogonal Curvilinear Coordinates:

If the metric is diagonal with scale factors $h_i$ such that $g_{ii} = h_i^2$, then $g^{ii} = \frac{1}{h_i^2}$ and $\sqrt{g} = h_1 h_2 \dots h_n$:

This single master formula reproduces the Laplacian in Cartesian, cylindrical, spherical polar, paraboloidal, and toroidal coordinates instantly!

1. Intrinsic / Absolute Derivative Along a Curve

Let $C: x^i = x^i(t)$ be a smooth parameterized curve on $M$, and let $v^i(t) = \frac{dx^i}{dt}$ be its tangent vector. If $A^i(t)$ is a vector field defined along the curve, its intrinsic (absolute) derivative $\frac{D A^i}{dt}$ is defined as the directional covariant derivative along the curve:

2. Definition of Parallel Transport

A vector field $A^i(t)$ is said to be parallel transported along the curve $C$ if its intrinsic derivative vanishes everywhere along the path:

Preservation of Norms and Angles:

Let $A^i$ and $B^i$ be parallel-transported along $C$. Using Ricci's Theorem:

Hence:

Parallel transport preserves both vector lengths and angles between vectors!

3. Geodesic Re-interpretation

A geodesic is a curve whose tangent vector $T^i = \frac{dx^i}{ds}$ is parallel transported along itself:

4. Holonomy Around a Closed Loop

In flat space, parallel transporting a vector around any closed loop returns it to its original orientation. In a curved space, parallel transport is path-dependent. Transporting a vector $A^i$ around an infinitesimal closed loop of area $\Delta \sigma^{jk}$ induces a shift:

The transformation matrix $A^i \to A^i + \Delta A^i$ is an element of the holonomy group $\text{Hol}(g) \subseteq O(n)$. Curvature is literally the infinitesimal generator of non-trivial holonomy!