Unit 3: Metric Tensors, Riemannian Spaces & Index Manipulation

1. The Riemannian Metric Axioms

In 1854, Bernhard Riemann generalized Euclidean geometry to arbitrary $n$-dimensional curved manifolds by postulating that the infinitesimal distance $ds$ between two neighboring points $P(x^i)$ and $Q(x^i + dx^i)$ is given by the square root of a quadratic differential form.

Definition 3.1 (Fundamental Metric Form): A differentiable manifold $V_n$ is a Riemannian space $R_n$ if it is equipped with a symmetric, second-rank covariant tensor field $g_{ij}(x) = g_{ji}(x)$ called the metric tensor, such that the arc length element $ds$ satisfies:

In a proper (positive-definite) Riemannian space, $ds^2 > 0$ for any non-zero displacement $dx^i \ne 0$.

Signature Classifications:

1. Positive-Definite Riemannian Space: All eigenvalues of the matrix $(g_{ij})$ are strictly positive ($++++\dots$). Distance $ds$ is always real and positive.

2. Pseudo-Riemannian (Lorentzian) Space: The metric is non-degenerate ($\det(g) \ne 0$) but indefinite. In 4D relativistic spacetime, the metric signature is $(-, +, +, +)$ (or $(+, -, -, -)$):

where intervals are classified into timelike ($ds^2 < 0$), spacelike ($ds^2 > 0$), and lightlike / null ($ds^2 = 0$).

2. Transformation Law of the Metric Tensor

Let $x^i \to \bar{x}^k$ be a regular coordinate transformation. Since the arc length $ds$ is an intrinsic physical distance, the quadratic form is an absolute scalar invariant:

Substituting the differential transformation $dx^i = \frac{\partial x^i}{\partial \bar{x}^k} d\bar{x}^k$ and $dx^j = \frac{\partial x^j}{\partial \bar{x}^l} d\bar{x}^l$:

Because the coordinate differentials $d\bar{x}^k$ are arbitrary, we deduce:

This rigorously confirms that $g_{ij}$ transforms as a symmetric covariant tensor of rank 2.

1. The Conjugate (Reciprocal) Metric Tensor $g^{ij}$

Let $g = \det(g_{ij})$ denote the determinant of the $n \times n$ matrix of metric components. For a non-degenerate Riemannian space, $g \ne 0$.

Definition 3.2 (Conjugate Metric Tensor): The conjugate (reciprocal) metric tensor $g^{ij}$ is defined as the matrix inverse of $g_{ij}$:

In terms of the matrix cofactors of $g_{ij}$:

where $G^{ij}$ is the cofactor of the entry $g_{ij}$ in the determinant $g$.

By symmetry of $g_{ij}$, the conjugate metric tensor is also symmetric: $g^{ij} = g^{ji}$.

2. Derivative Identities of the Metric Determinant

The derivative of the metric determinant with respect to coordinate $x^k$ plays a central role in constructing Christoffel symbols and invariant divergence formulas.

Theorem 3.1 (Derivative of Metric Determinant): Let $g = \det(g_{ij})$. Then:

or equivalently:

Proof: Expanding determinant $g$ along its $i$-th row:

Differentiating with respect to $g_{ij}$ yields $\frac{\partial g}{\partial g_{ij}} = G^{ij} = g \, g^{ij}$. By the multivariable chain rule:

Similarly, differentiating the relation $g^{ij} g_{jk} = \delta_k^i$:

1. Natural Tangent Basis Vectors $\mathbf{e}_i$

Let $\mathbf{r}(x^1, \dots, x^n)$ be the position vector in an embedding space. The natural coordinate basis vectors tangent to the coordinate curves are:

The infinitesimal displacement vector is:

Computing the squared arc length:

Comparing directly with $ds^2 = g_{ij} dx^i dx^j$, we obtain the profound geometric identity:

The components of the metric tensor are the scalar products of the natural tangent basis vectors!

2. Reciprocal Dual Basis Vectors $\mathbf{e}^i$

Definition 3.3 (Dual Reciprocal Basis): The reciprocal basis vectors $\mathbf{e}^1, \mathbf{e}^2, \dots, \mathbf{e}^n$ are the unique vectors satisfying the biorthogonality condition:

In terms of the metric tensor and conjugate metric:

Computing their mutual inner products:

Thus, the conjugate metric components are the scalar products of the reciprocal dual basis vectors:

1. The Musical Isomorphisms ($\flat$ and $\sharp$)

The metric tensor establishes a canonical linear isomorphism between the tangent space $T_p M$ (contravariant vectors) and cotangent space $T_p^* M$ (covariant covectors). In differential geometry, these are called the musical isomorphisms:

• Flat ($\flat$, Index Lowering): Converts a contravariant vector into a covariant covector.
• Sharp ($\sharp$, Index Raising): Converts a covariant covector into a contravariant vector.

2. Algebraic Index Manipulation

For any contravariant vector $A^i$:

Conversely, for any covariant vector $A_i$:

Consistency verification using the inverse relationship:

The operations of raising and lowering indices are exact mutual inverses!

Raising and Lowering in Higher-Rank Tensors:

The metric tensor raises and lowers individual indices independently:

• Lowering the first index of $T^{ij}$:

• Lowering both indices of $T^{ij}$:

• Mixed associated tensor from covariant tensor $T_{ij}$:

(Note: If $T_{ij}$ is not symmetric, the position of the dot placeholder matters: $T^i_{\; j} \ne T_j^{\; i}$).

1. Vector Magnitude and Angle between Vectors

In a Riemannian space with metric $g_{ij}$:

1. Magnitude (Norm) of a Vector:

The length $\|A\|$ of a contravariant vector $A^i$ is defined by:

For a covariant vector $B_i$:

2. Angle between Two Vectors:

The angle $\theta$ between two non-zero contravariant vectors $A^i$ and $B^i$ is:

By the Cauchy-Schwarz inequality for positive-definite metrics, $-1 \le \cos\theta \le 1$.

3. Orthogonality Condition:

Two vectors $A^i$ and $B^i$ are orthogonal if and only if their Riemannian inner product vanishes:

2. Orthogonal Curvilinear Coordinate Systems

A coordinate system is orthogonal if the coordinate lines intersect at right angles everywhere. In an orthogonal coordinate system:

The metric tensor and its conjugate are purely diagonal:

The metric determinant is simply the product of diagonal entries:

where $h_i = \sqrt{g_{ii}}$ are the Lamé scale factors.