Riemannian manifold

In differential geometry, a Riemannian manifold (or Riemann space) is a geometric space on which many geometric notions such as distance, angles, length, volume, and curvature are defined. Euclidean space, the \(n\)-sphere, hyperbolic space, and smooth surfaces in three-dimensional space, such as ellipsoids and paraboloids, are all examples of Riemannian manifolds. Riemannian manifolds take their name from German mathematician Bernhard Riemann, who first conceptualized them in 1854.

Formally, a Riemannian metric (or just a metric) on a smooth manifold is a smoothly varying choice of inner product for each tangent space of the manifold. A Riemannian manifold is a smooth manifold together with a Riemannian metric. The techniques of differential and integral calculus are used to pull geometric data out of the Riemannian metric. For example, integration leads to the Riemannian distance function, whereas differentiation is used to define curvature and parallel transport.

Any smooth surface in three-dimensional Euclidean space is a Riemannian manifold with a Riemannian metric coming from the way it sits inside the ambient space. The same is true for any submanifold of Euclidean space of any dimension. Although John Nash proved that every Riemannian manifold arises as a submanifold of Euclidean space, and although some Riemannian manifolds are naturally exhibited or defined in that way, the idea of a Riemannian manifold emphasizes the intrinsic point of view, which defines geometric notions directly on the abstract space itself without referencing an ambient space. In many instances, such as for hyperbolic space and projective space, Riemannian metrics are more naturally defined or constructed using the intrinsic point of view. Additionally, many metrics on Lie groups and homogeneous spaces are defined intrinsically by using group actions to transport an inner product on a single tangent space to the entire manifold, and many special metrics such as constant scalar curvature metrics and Kähler-Einstein metrics are constructed intrinsically using tools from partial differential equations.

Riemannian geometry, the study of Riemannian manifolds, has deep connections to other areas of mathematics, including geometric topology, complex geometry, and algebraic geometry. Applications include physics (especially general relativity and gauge theory), computer graphics, machine learning, and cartography. Generalizations of Riemannian manifolds include pseudo-Riemannian manifolds, Finsler manifolds, and sub-Riemannian manifolds.

In 1827, Carl Friedrich Gauss discovered that the Gaussian curvature of a surface embedded in 3-dimensional space only depends on local measurements made within the surface (the first fundamental form). This result is known as the Theorema Egregium ("remarkable theorem" in Latin).

A map that preserves the local measurements of a surface is called a local isometry. A property of a surface is called an intrinsic property if it is preserved by local isometries and it is called an extrinsic property if it is not. In this language, the Theorema Egregium says that the Gaussian curvature is an intrinsic property of surfaces.

Riemannian manifolds and their curvature were first introduced non-rigorously by Bernhard Riemann in 1854. However, they would not be formalized until much later. In fact, the more primitive concept of a smooth manifold was first explicitly defined only in 1913 in a book by Hermann Weyl.

Élie Cartan introduced the Cartan connection, one of the first concepts of a connection. Levi-Civita defined the Levi-Civita connection, a special connection on a Riemannian manifold.

Albert Einstein used the theory of pseudo-Riemannian manifolds (a generalization of Riemannian manifolds) to develop general relativity. Specifically, the Einstein field equations are constraints on the curvature of spacetime, which is a 4-dimensional pseudo-Riemannian manifold.

Let \(M\) be a smooth manifold. For each point \(p \in M\), there is an associated vector space \(T_pM\) called the tangent space of \(M\) at \(p\). Vectors in \(T_pM\) are thought of as the vectors tangent to \(M\) at \(p\).

However, \(T_pM\) does not come equipped with an inner product, a "measuring stick" that gives tangent vectors a concept of length and angle. This is an important deficiency because calculus teaches that to calculate the length of a curve, the length of vectors tangent to the curve must be defined. A Riemannian metric puts such a "measuring stick" on every tangent space.

A Riemannian metric \(g\) on \(M\) assigns to each \(p\) a positive-definite symmetric bilinear form (i.e. an inner product) \(g_p : T_pM \times T_pM \to \mathbb R\) in a smooth way (see the section on regularity below). This induces a norm \(\|\cdot\|_p : T_pM \to \mathbb R\) defined by \(\|v\|_p = \sqrt{g_p(v,v)}\). A smooth manifold \(M\) endowed with a Riemannian metric \(g\) is a Riemannian manifold, denoted \((M,g)\). A Riemannian metric is a special case of a metric tensor.

A Riemannian metric is not to be confused with the distance function of a metric space, which is also called a metric.

An isometry is a function between Riemannian manifolds which preserves all of the structure of Riemannian manifolds. If two Riemannian manifolds have an isometry between them, they are called isometric, and they are considered to be the same manifold for the purpose of Riemannian geometry.

Specifically, if \((M,g)\) and \((N,h)\) are two Riemannian manifolds, a diffeomorphism \(f:M\to N\) is called an isometry if \(g=f^\ast h\), that is, if

\(g_p(u,v)=h_{f(p)}(df_p(u),df_p(v))\)

for all \(p\in M\) and \(u,v\in T_pM.\) For example, translations and rotations are both isometries from Euclidean space (to be defined soon) to itself.

One says that a smooth map \(f:M\to N,\) not assumed to be a diffeomorphism, is a local isometry if every \(p\in M\) has an open neighborhood \(U\) such that \(f:U\to f(U)\) is an isometry (and thus a diffeomorphism).

An oriented \(n\)-dimensional Riemannian manifold \((M,g)\) has a unique \(n\)-form \(dV_g\) called the Riemannian volume form. The Riemannian volume form is preserved by orientation-preserving isometries. The volume form gives rise to a measure on \(M\) which allows measurable functions to be integrated. If \(M\) is compact, the volume of \(M\) is \(\int_M dV_g\).

Let \(x^1,\ldots,x^n\) denote the standard coordinates on \(\mathbb{R}^n.\) The (canonical) Euclidean metric \(g^\text{can}\) is given by

\(g^\text{can}\left(\sum_i a_i \frac{\partial}{\partial x^i}, \sum_j b_j \frac{\partial}{\partial x^j} \right) = \sum_i a_i b_i\)

or equivalently

\(g^\text{can} = (dx^1)^2 + \cdots + (dx^n)^2\)

or equivalently by its coordinate functions

\(g_{ij}^\text{can} = \delta_{ij}\) where \(\delta_{ij}\) is the Kronecker delta

which together form the matrix

\((g_{ij}^\text{can}) = \begin{pmatrix} 1 & 0 & \cdots & 0 \\ 0 & 1 & \cdots & 0 \\ \vdots & \vdots & \ddots & \vdots \\ 0 & 0 & \cdots & 1 \end{pmatrix}.\)

The Riemannian manifold \((\mathbb{R}^n,g^\text{can})\) is called Euclidean space.

Let \((M,g)\) be a Riemannian manifold and let \(i : N \to M\) be an immersed submanifold or an embedded submanifold of \(M\). The pullback \(i^*g\) of \(g\) is a Riemannian metric on \(N\), called the induced metric, and \((N, i^*g)\) is said to be a Riemannian submanifold of \((M,g)\).

In the case where \(N \subseteq M\), the map \(i : N \to M\) is given by \(i(x) = x\) and the metric \(i^*g\) is just the restriction of \(g\) to vectors tangent along \(N\). In general, the formula for \(i^*g\) is

\(i^*g_p(v,w) = g_{i(p)} \big( di_p(v), di_p(w) \big),\)

where \(di_p(v)\) is the pushforward of \(v\) by \(i.\)

Examples:

• The \(n\)-sphere

  \(S^n=\{x\in\mathbb{R}^{n+1}:(x^1)^2+\cdots+(x^{n+1})^2=1\}\)

is a smooth embedded submanifold of Euclidean space \(\mathbb R^{n+1}\). The Riemannian metric this induces on \(S^n\) is called the round metric or standard metric.

• Fix real numbers \(a,b,c\). The ellipsoid

  \(\left\{(x,y,z) \in \mathbb R^3 : \frac{x^2}{a^2} + \frac{y^2}{b^2} + \frac{z^2}{c^2} = 1 \right\}\)

is a smooth embedded submanifold of Euclidean space \(\mathbb R^3\).

• The graph of a smooth function \(f:\mathbb{R}^n\to\mathbb{R}\) is a smooth embedded submanifold of \(\mathbb{R}^{n+1}\) with its standard metric.
• If \((M,g)\) is not simply connected, there is a covering map \(\widetilde{M}\to M\), where \(\widetilde M\) is the universal cover of \(M\). This is an immersion (since it is locally a diffeomorphism), so \(\widetilde M\) automatically inherits a Riemannian metric. By the same principle, any smooth covering space of a Riemannian manifold inherits a Riemannian metric.

On the other hand, if \(N\) already has a Riemannian metric \(\tilde g\), then the immersion (or embedding) \(i : N \to M\) is called an isometric immersion (or isometric embedding) if \(\tilde g = i^* g\). Hence isometric immersions and isometric embeddings are Riemannian submanifolds.

Let \((M,g)\) and \((N,h)\) be two Riemannian manifolds, and consider the product manifold \(M\times N\). The Riemannian metrics \(g\) and \(h\) naturally put a Riemannian metric \(\widetilde{g}\) on \(M\times N,\) which can be described in a few ways.

• Considering the decomposition \(T_{(p,q)}(M\times N) \cong T_pM \oplus T_qN,\) one may define

  \(\widetilde{g}_{p,q} ((u_1, u_2), (v_1, v_2)) = g_p(u_1, v_1) + h_q(u_2, v_2).\)

• If \((U,x)\) is a smooth coordinate chart on \(M\) and \((V,y)\) is a smooth coordinate chart on \(N\), then \((U \times V, (x,y))\) is a smooth coordinate chart on \(M \times N.\) Let \(g_U\) be the representation of \(g\) in the chart \((U,x)\) and let \(h_V\) be the representation of \(h\) in the chart \((V,y)\). The representation of \(\widetilde{g}\) in the coordinates \((U \times V,(x,y))\) is

  \(\widetilde{g} = \sum_{ij} \widetilde{g}_{ij} \, dx^i \, dx^j \text{ where } (\widetilde{g}_{ij}) = \begin{pmatrix} g_U & 0 \\ 0 & h_V \end{pmatrix}.\)

For example, consider the \(n\)-torus \(T^n = S^1\times\cdots\times S^1\). If each copy of \(S^1\) is given the round metric, the product Riemannian manifold \(T^n\) is called the flat torus. As another example, the Riemannian product \(\mathbb R \times \cdots \times \mathbb R\), where each copy of \(\mathbb R\) has the Euclidean metric, is isometric to \(\mathbb R^n\) with the Euclidean metric.

Let \(g_1, \ldots, g_k\) be Riemannian metrics on \(M.\) If \(f_1, \ldots, f_k\) are any positive smooth functions on \(M\), then \(f_1 g_1 + \ldots + f_k g_k\) is another Riemannian metric on \(M.\)

Theorem: Every smooth manifold admits a (non-canonical) Riemannian metric.

This is a fundamental result. Although much of the basic theory of Riemannian metrics can be developed using only that a smooth manifold is a locally Euclidean topological space, for this result it is necessary to use that smooth manifolds are Hausdorff and paracompact. The reason is that the proof makes use of a partition of unity.

An alternative proof uses the Whitney embedding theorem to embed \(M\) into Euclidean space and then pulls back the metric from Euclidean space to \(M\). On the other hand, the Nash embedding theorem states that, given any smooth Riemannian manifold \((M,g),\) there is an embedding \(F:M\to\mathbb{R}^N\) for some \(N\) such that the pullback by \(F\) of the standard Riemannian metric on \(\mathbb{R}^N\) is \(g.\) That is, the entire structure of a smooth Riemannian manifold can be encoded by a diffeomorphism to a certain embedded submanifold of some Euclidean space. Therefore, one could argue that nothing can be gained from the consideration of abstract smooth manifolds and their Riemannian metrics. However, there are many natural smooth Riemannian manifolds, such as the set of rotations of three-dimensional space and hyperbolic space, of which any representation as a submanifold of Euclidean space will fail to represent their remarkable symmetries and properties as clearly as their abstract presentations do.

An admissible curve is a piecewise smooth curve \(\gamma : [0,1] \to M\) whose velocity \(\gamma'(t) \in T_{\gamma(t)}M\) is nonzero everywhere it is defined. The nonnegative function \(t\mapsto\|\gamma'(t)\|_{\gamma(t)}\) is defined on the interval \([0,1]\) except for at finitely many points. The length \(L(\gamma)\) of an admissible curve \(\gamma : [0,1] \to M\) is defined as

\(L(\gamma)=\int_0^1 \|\gamma'(t)\|_{\gamma(t)} \, dt.\)

The integrand is bounded and continuous except at finitely many points, so it is integrable. For \((M,g)\) a connected Riemannian manifold, define \(d_g:M\times M\to[0,\infty)\) by

\(d_g(p,q) = \inf \{ L(\gamma) : \gamma \text{ an admissible curve with } \gamma(0) = p, \gamma(1) = q \}.\)

Theorem: \((M,d_g)\) is a metric space, and the metric topology on \((M,d_g)\) coincides with the topology on \(M\).

Although the length of a curve is given by an explicit formula, it is generally impossible to write out the distance function \(d_g\) by any explicit means. In fact, if \(M\) is compact, there always exist points where \(d_g:M\times M\to\mathbb{R}\) is non-differentiable, and it can be remarkably difficult to even determine the location or nature of these points, even in seemingly simple cases such as when \((M,g)\) is an ellipsoid.

If one works with Riemannian metrics that are merely continuous but possibly not smooth, the length of an admissible curve and the Riemannian distance function are defined exactly the same, and, as before, \((M,d_g)\) is a metric space and the metric topology on \((M,d_g)\) coincides with the topology on \(M\).

The diameter of the metric space \((M,d_g)\) is

\(\operatorname{diam}(M,d_g)=\sup\{d_g(p,q):p,q\in M\}.\)

The Hopf-Rinow theorem shows that if \((M,d_g)\) is complete and has finite diameter, it is compact. Conversely, if \((M,d_g)\) is compact, then the function \(d_g:M\times M\to\mathbb{R}\) has a maximum, since it is a continuous function on a compact metric space. This proves the following.

If \((M,d_g)\) is complete, then it is compact if and only if it has finite diameter.

This is not the case without the completeness assumption; for counterexamples one could consider any open bounded subset of a Euclidean space with the standard Riemannian metric. It is also not true that any complete metric space of finite diameter must be compact; it matters that the metric space came from a Riemannian manifold.

An (affine) connection is an additional structure on a Riemannian manifold that defines differentiation of one vector field with respect to another. Connections contain geometric data, and two Riemannian manifolds with different connections have different geometry.

Let \(\mathfrak X(M)\) denote the space of vector fields on \(M\). An (affine) connection

\(\nabla : \mathfrak X(M) \times \mathfrak X(M) \to \mathfrak X(M)\)

on \(M\) is a bilinear map \((X,Y) \mapsto \nabla_X Y\) such that

1. For every function \(f \in C^\infty(M)\), \(\nabla_{f_1 X_1 + f_2 X_2} Y = f_1 \,\nabla_{X_1} Y + f_2 \, \nabla_{X_2} Y,\)
2. The product rule \(\nabla_X fY=X(f)Y+ f\,\nabla_X Y\) holds.

The expression \(\nabla_X Y\) is called the covariant derivative of \(Y\) with respect to \(X\).

Two Riemannian manifolds with different connections have different geometry. Thankfully, there is a natural connection associated to a Riemannian manifold called the Levi-Civita connection.

A connection \(\nabla\) is said to preserve the metric if

\(X\bigl(g(Y,Z)\bigr) = g(\nabla_X Y, Z) + g(Y, \nabla_X Z)\)

A connection \(\nabla\) is torsion-free if

\(\nabla_X Y - \nabla_Y X = [X,Y],\)

where \([\cdot,\cdot]\) is the Lie bracket.

A Levi-Civita connection is a torsion-free connection that preserves the metric. Once a Riemannian metric is fixed, there exists a unique Levi-Civita connection. Note that the definition of preserving the metric uses the regularity of \(g\).