Riemannian metrics are the heart of Riemannian geometry, giving curved spaces a way to measure distances and angles. They define inner products on tangent spaces, allowing us to quantify geometric properties on manifolds.

These metrics have key features: positive definiteness, smoothness, and symmetry. These properties ensure we can measure lengths, differentiate quantities, and maintain consistency in our calculations on curved spaces.

Riemannian Metric Properties

Fundamental Characteristics of Riemannian Metrics

• Riemannian metric defines a smooth inner product on the tangent space of each point in a manifold
• Positive definiteness ensures the inner product is always positive for non-zero vectors
• Smoothness requires the metric to vary smoothly across the manifold
• Symmetry property guarantees the inner product is symmetric for any pair of vectors

Mathematical Formulation and Implications

• Riemannian metric denoted as $g$ or $\langle \cdot, \cdot \rangle$, assigns an inner product to each tangent space
• Positive definiteness expressed as $g(v,v) > 0$ for all non-zero tangent vectors $v$
• Smoothness implies the metric components are smooth functions in local coordinates
• Symmetry formulated as $g(u,v) = g(v,u)$ for any tangent vectors $u$ and $v$

Applications and Geometric Interpretations

• Riemannian metrics enable measurement of distances, angles, and volumes on curved spaces
• Positive definiteness allows definition of length and ensures non-degenerate geometry
• Smoothness permits differentiation and integration of metric-dependent quantities
• Symmetry simplifies calculations and ensures consistency in geometric measurements

Metric Tensor and Inner Product

Metric Tensor Fundamentals

• Metric tensor represents the Riemannian metric in local coordinates
• Components of the metric tensor denoted as $g_{ij}$ in a coordinate basis
• Metric tensor transforms as a $(0,2)$-tensor under coordinate changes
• Inverse metric tensor $g^{ij}$ exists due to positive definiteness

Inner Product Structure and Properties

• Inner product on tangent spaces defined by the metric tensor
• For tangent vectors $v = v^i \frac{\partial}{\partial x^i}$ and $w = w^j \frac{\partial}{\partial x^j}$, inner product given by $g(v,w) = g_{ij} v^i w^j$
• Inner product satisfies linearity, symmetry, and positive definiteness
• Allows computation of vector magnitudes and angles between vectors

Tangent Space and Geometric Significance

• Tangent space at a point consists of all tangent vectors at that point
• Metric tensor provides a way to measure in the tangent space
• Tangent space equipped with inner product becomes an inner product space
• Geometric quantities (lengths, angles) in tangent space relate to those on the manifold

Coordinate Representation and Pullback

Coordinate Representation of Riemannian Metrics

• In local coordinates $(x^1, \ldots, x^n)$, metric tensor represented by matrix $(g_{ij})$
• Components $g_{ij}$ are functions of the coordinates
• Metric can be written as $g = g_{ij} dx^i \otimes dx^j$ using the exterior product
• Coordinate representation allows explicit calculations and transformations

Pullback Metric and Induced Metrics

• Pullback metric arises when mapping between manifolds
• For a smooth map $f: M \to N$ between manifolds, pullback metric $f^g$ on $M$ induced by metric $g$ on $N$
• Pullback metric defined as $(f^g)p(v,w) = g{f(p)}(df_p(v), df_p(w))$ for tangent vectors $v,w$ at $p \in M$
• Allows study of geometric properties of submanifolds and embedded surfaces

Transformations and Coordinate Changes

• Under coordinate change, metric components transform according to $g_{ij} = \frac{\partial y^a}{\partial x^i} \frac{\partial y^b}{\partial x^j} \tilde{g}_{ab}$
• Pullback metric components in new coordinates given by $(\tilde{f^g}){ij} = \frac{\partial f^a}{\partial x^i} \frac{\partial f^b}{\partial x^j} g{ab}$
• Coordinate representations and transformations crucial for practical computations in Riemannian geometry