3.6 Warped product metrics

Warped product metrics are a fascinating generalization of direct product metrics. They allow for the scaling of the fiber manifold's metric by a warping function dependent on the base manifold, leading to rich geometric structures with diverse applications.

These metrics offer unique insights into curvature, geodesics, and isometries. By studying warped products, we gain a deeper understanding of manifold geometry and its connections to physics, particularly in areas like general relativity and cosmology.

Definition of warped product metrics

• Warped product metrics are a type of metric structure on a product manifold that generalizes the notion of a direct product metric
• They allow for the scaling of the metric on the fiber manifold by a warping function that depends on the point in the base manifold
• Warped product metrics have important applications in various areas of differential geometry and mathematical physics, such as general relativity and the study of Riemannian submersions

Warping functions

• A warping function is a smooth, positive function $f: B \to \mathbb{R}^+$ defined on the base manifold $B$
• The warping function determines how the metric on the fiber manifold $F$ is scaled at each point of the base manifold
• The metric on the warped product manifold $B \times_f F$ is given by $g = g_B + f^2 g_F$, where $g_B$ and $g_F$ are the metrics on the base and fiber manifolds, respectively
• The choice of the warping function significantly influences the geometric properties of the warped product manifold, such as its curvature and geodesics

Base manifolds and fiber manifolds

• In a warped product manifold $B \times_f F$, the manifold $B$ is called the base manifold, and the manifold $F$ is called the fiber manifold
• The base manifold and fiber manifold are Riemannian manifolds with their own metrics, denoted by $g_B$ and $g_F$, respectively
• The dimensions of the base manifold and fiber manifold are denoted by $\dim B = m$ and $\dim F = k$, and the dimension of the warped product manifold is $m + k$
• The topology of the warped product manifold is the product topology, meaning that open sets in $B \times_f F$ are products of open sets in $B$ and $F$

Curvature of warped products

• The curvature of a warped product manifold depends on the curvatures of the base manifold and fiber manifold, as well as the warping function
• The Riemann curvature tensor, Ricci tensor, and scalar curvature of a warped product can be expressed in terms of the corresponding quantities on the base and fiber manifolds and the warping function
• The curvature of a warped product manifold can exhibit interesting properties, such as the presence of singularities or the existence of Einstein metrics

Ricci curvature

• The Ricci curvature of a warped product manifold $B \times_f F$ can be expressed in terms of the Ricci curvatures of the base manifold $B$ and the fiber manifold $F$, as well as the warping function $f$ and its derivatives
• For a vector $X$ tangent to the base manifold, the Ricci curvature is given by $\mathrm{Ric}(X, X) = \mathrm{Ric}_B(X, X) - k \frac{\nabla^2 f(X, X)}{f}$, where $\mathrm{Ric}_B$ is the Ricci curvature of the base manifold and $k$ is the dimension of the fiber manifold
• For a vector $Y$ tangent to the fiber manifold, the Ricci curvature is given by $\mathrm{Ric}(Y, Y) = \mathrm{Ric}_F(Y, Y) - (f \Delta f + (m-1) |\nabla f|^2) g_F(Y, Y)$, where $\mathrm{Ric}_F$ is the Ricci curvature of the fiber manifold, $\Delta f$ is the Laplacian of $f$, and $m$ is the dimension of the base manifold

Scalar curvature

• The scalar curvature of a warped product manifold $B \times_f F$ can be expressed in terms of the scalar curvatures of the base manifold $B$ and the fiber manifold $F$, as well as the warping function $f$ and its derivatives
• The scalar curvature is given by $S = S_B + \frac{S_F}{f^2} - \frac{2k}{f} \Delta f - k(k-1) \frac{|\nabla f|^2}{f^2}$, where $S_B$ and $S_F$ are the scalar curvatures of the base manifold and fiber manifold, respectively, and $\Delta f$ is the Laplacian of $f$
• The scalar curvature of a warped product manifold can be used to study the global geometry of the manifold, such as the existence of Einstein metrics or the behavior of the manifold at infinity

Sectional curvature

• The sectional curvature of a warped product manifold $B \times_f F$ depends on the sectional curvatures of the base manifold $B$ and the fiber manifold $F$, as well as the warping function $f$ and its derivatives
• For a plane spanned by orthonormal vectors $X$ and $Y$ tangent to the base manifold, the sectional curvature is given by $K(X, Y) = K_B(X, Y)$, where $K_B$ is the sectional curvature of the base manifold
• For a plane spanned by orthonormal vectors $V$ and $W$ tangent to the fiber manifold, the sectional curvature is given by $K(V, W) = \frac{K_F(V, W) - |\nabla f|^2}{f^2}$, where $K_F$ is the sectional curvature of the fiber manifold
• For a plane spanned by a unit vector $X$ tangent to the base manifold and a unit vector $V$ tangent to the fiber manifold, the sectional curvature is given by $K(X, V) = -\frac{\nabla^2 f(X, X)}{f}$

Geodesics in warped products

• Geodesics in a warped product manifold $B \times_f F$ are curves that minimize the distance between points in the manifold
• The behavior of geodesics in a warped product manifold is influenced by the warping function and the geodesics of the base manifold and fiber manifold
• Studying geodesics in warped product manifolds is important for understanding the global geometry of the manifold and its implications in various applications, such as general relativity

Clairaut's theorem

• Clairaut's theorem is a result that describes the behavior of geodesics in warped product manifolds with a warping function that depends only on the distance from a fixed point in the base manifold
• In such manifolds, called rotationally symmetric manifolds, Clairaut's theorem states that the quantity $r \sin \alpha$ is constant along a geodesic, where $r$ is the distance from the fixed point and $\alpha$ is the angle between the geodesic and the radial direction
• Clairaut's theorem provides a useful tool for studying the qualitative behavior of geodesics in rotationally symmetric warped product manifolds, such as surfaces of revolution

Geodesic equations

• The geodesic equations in a warped product manifold $B \times_f F$ can be derived using the Christoffel symbols of the warped product metric
• For a curve $\gamma(t) = (x(t), y(t))$ in $B \times_f F$, where $x(t)$ is a curve in the base manifold and $y(t)$ is a curve in the fiber manifold, the geodesic equations are:
  • $\ddot{x}^i + \Gamma^i_{jk} \dot{x}^j \dot{x}^k + 2 \frac{f'}{f} \dot{x}^i |\dot{y}|^2 = 0$
  • $\ddot{y}^{\alpha} + \tilde{\Gamma}^{\alpha}_{\beta \gamma} \dot{y}^{\beta} \dot{y}^{\gamma} - \frac{f'}{f} |\dot{x}|^2 y^{\alpha} = 0$
• Here, $\Gamma^i_{jk}$ are the Christoffel symbols of the base manifold, $\tilde{\Gamma}^{\alpha}_{\beta \gamma}$ are the Christoffel symbols of the fiber manifold, and $f'$ denotes the derivative of the warping function along the curve $x(t)$

Geodesic completeness

• A Riemannian manifold is said to be geodesically complete if every geodesic can be extended indefinitely in both directions
• The geodesic completeness of a warped product manifold $B \times_f F$ depends on the geodesic completeness of the base manifold $B$ and the fiber manifold $F$, as well as the behavior of the warping function $f$
• If the base manifold $B$ and the fiber manifold $F$ are both geodesically complete and the warping function $f$ is bounded away from zero, then the warped product manifold $B \times_f F$ is geodesically complete
• However, if either the base manifold or the fiber manifold is not geodesically complete, or if the warping function approaches zero at some point, the warped product manifold may not be geodesically complete

Isometries of warped products

• An isometry is a distance-preserving map between Riemannian manifolds
• The isometry group of a warped product manifold $B \times_f F$ is related to the isometry groups of the base manifold $B$ and the fiber manifold $F$, as well as the symmetries of the warping function $f$
• Understanding the isometries of warped product manifolds is important for studying the symmetries and the global geometry of these manifolds

Vertical isometries

• A vertical isometry of a warped product manifold $B \times_f F$ is an isometry that preserves the fibers $\{x\} \times F$ for each point $x$ in the base manifold $B$
• The vertical isometries of a warped product manifold form a subgroup of the isometry group, called the vertical isometry group
• The vertical isometry group of a warped product manifold is isomorphic to the isometry group of the fiber manifold $F$
• Vertical isometries play a crucial role in understanding the symmetries of the fiber manifold and their impact on the geometry of the warped product manifold

Horizontal isometries

• A horizontal isometry of a warped product manifold $B \times_f F$ is an isometry that preserves the leaves $B \times \{y\}$ for each point $y$ in the fiber manifold $F$
• The horizontal isometries of a warped product manifold form a subgroup of the isometry group, called the horizontal isometry group
• The horizontal isometry group of a warped product manifold is related to the isometry group of the base manifold $B$ and the symmetries of the warping function $f$
• In some cases, such as when the warping function is constant, the horizontal isometry group of a warped product manifold can be isomorphic to the isometry group of the base manifold

Warped product isometry groups

• The isometry group of a warped product manifold $B \times_f F$ is a subgroup of the product of the isometry groups of the base manifold $B$ and the fiber manifold $F$
• The structure of the warped product isometry group depends on the compatibility of the isometries of the base manifold and the fiber manifold with the warping function $f$
• In some cases, the warped product isometry group can be expressed as a semidirect product of the horizontal isometry group and the vertical isometry group
• The warped product isometry group provides valuable insights into the symmetries and the global geometry of the warped product manifold, which are essential for various applications in differential geometry and mathematical physics

Examples of warped products

• Warped product manifolds arise naturally in various contexts in differential geometry and mathematical physics
• Some notable examples of warped product manifolds include surfaces of revolution, generalized cylinders, and Robertson-Walker spacetimes
• Studying these examples helps to illustrate the geometric properties of warped product manifolds and their applications in different areas of mathematics and physics

Surfaces of revolution

• A surface of revolution is a warped product manifold obtained by rotating a curve in the $xy$-plane around the $x$-axis
• The base manifold is an interval $I \subset \mathbb{R}$, and the fiber manifold is a circle $S^1$
• The warping function $f: I \to \mathbb{R}^+$ is the distance from the curve to the $x$-axis
• The metric on a surface of revolution is given by $ds^2 = dx^2 + f(x)^2 d\theta^2$, where $x$ is the parameter along the base interval and $\theta$ is the angular coordinate on the circle
• Examples of surfaces of revolution include the sphere, the torus, and the paraboloid

Generalized cylinders

• A generalized cylinder is a warped product manifold obtained by taking the product of a curve in a Riemannian manifold $M$ with another Riemannian manifold $N$
• The base manifold is the curve $\gamma: I \to M$, and the fiber manifold is the Riemannian manifold $N$
• The warping function $f: I \to \mathbb{R}^+$ is a positive function along the curve
• The metric on a generalized cylinder is given by $ds^2 = dt^2 + f(t)^2 g_N$, where $t$ is the parameter along the base curve and $g_N$ is the metric on the fiber manifold
• Generalized cylinders can be used to model various geometric objects, such as ruled surfaces and warped products of higher-dimensional manifolds

Robertson-Walker spacetimes

• Robertson-Walker spacetimes are warped product manifolds that model homogeneous and isotropic universes in general relativity
• The base manifold is an interval $I \subset \mathbb{R}$ representing cosmic time, and the fiber manifold is a three-dimensional space of constant curvature (either spherical, Euclidean, or hyperbolic)
• The warping function $a: I \to \mathbb{R}^+$ is called the scale factor and describes the expansion or contraction of the universe over time
• The metric on a Robertson-Walker spacetime is given by $ds^2 = -dt^2 + a(t)^2 g_k$, where $g_k$ is the metric on the space of constant curvature with curvature $k \in \{1, 0, -1\}$
• Robertson-Walker spacetimes are the foundation of the standard cosmological models, such as the Big Bang model and the inflationary universe model

Comparison with direct products

• Direct product manifolds are a special case of warped product manifolds where the warping function is constant
• Comparing the properties of warped product manifolds with those of direct product manifolds helps to highlight the effects of the warping function on the geometry of the manifold
• Understanding the similarities and differences between warped products and direct products is important for various applications in differential geometry and mathematical physics

Metric properties

• In a direct product manifold $M \times N$, the metric is given by the sum of the metrics on the factor manifolds: $g = g_M + g_N$
• In a warped product manifold $B \times_f F$, the metric is given by $g = g_B + f^2 g_F$, where the metric on the fiber manifold is scaled by the square of the warping function
• The presence of the warping function in the metric of a warped product manifold leads to different geometric properties compared to direct product manifolds, such as the behavior of distances, angles, and curvatures

Curvature properties

• The curvature of a direct product manifold $M \times N$ is completely determined by the curvatures of the factor manifolds $M$ and $N$
• In a warped product manifold $B \times_f F$, the curvature depends on the curvatures of the base manifold $B$ and the fiber manifold $F$, as well as the warping function $f$ and its derivatives
• The presence of the warping function in the curvature expressions of a warped product manifold can lead to interesting geometric phenomena, such as the existence of singularities or the realization of Einstein metrics, which are not possible in direct product manifolds

Isometry groups

• The isometry group of a direct product manifold $M \times N$ is the product of the isometry groups of the factor manifolds: $\mathrm{Isom}(M \times N) = \mathrm{Isom}(M) \times \mathrm{Isom}(N)$
• In a warped product manifold $B \times_f F$, the isometry group is