List of formulas in Riemannian geometry

This is a list of important formulas used in Riemannian geometry, a part of mathematics that studies the shapes and properties of curved spaces. Riemannian geometry helps us understand how to measure distances, angles, and areas on surfaces that are not flat, like the curved surface of a sphere or a complicated 3D shape. Many of these formulas are used in physics, especially in the theory of gravity and the study of space and time.

The article uses Einstein notation, a special way of writing equations that makes them shorter and easier to work with in calculations. This notation is very common in advanced math and physics. You’ll see many symbols and equations here, and they might look complicated at first, but they are tools that mathematicians and scientists use to describe the world around us in a precise way.

Riemannian geometry was developed by the mathematician Bernhard Riemann in the 19th century, and it has become a foundation for many modern theories. From understanding the curves of the Earth to the bending of light in space, these formulas help us describe the geometry of the universe in ways that flat, everyday geometry cannot.

Christoffel symbols, covariant derivative

In Riemannian geometry, Christoffel symbols are important tools for describing how vectors change as we move on a curved surface. They help us understand the geometry of spaces that are not flat.

The Christoffel symbols can be of two kinds, and they are connected to the metric tensor, which measures distances on the surface. The covariant derivative is a way to take derivatives of vectors and tensors that accounts for the curvature of the space. This helps us study the motion of particles and the behavior of physical laws on curved surfaces.

Curvature tensors

Curvature tensors are important tools in Riemannian geometry, which is a branch of mathematics that studies the shapes of curved spaces. These tensors help describe how space curves and bends.

The most fundamental curvature tensor is the Riemann curvature tensor. It captures the intrinsic curvature of a space by measuring how vectors change when parallel transported along curves. Other important tensors derived from it include the Ricci curvature, which simplifies the Riemann tensor to a more manageable form, and the scalar curvature, which further reduces the Ricci tensor to a single number that describes the overall curvature of the space. These tensors are essential for understanding the geometry of surfaces and higher-dimensional spaces.

Gradient, divergence, Laplace–Beltrami operator

The gradient of a function shows how the function changes at each point. It is found by looking at the small changes in the function’s value in every direction.

Divergence measures how much a vector field spreads out from a point. It is calculated using the changes in the vector field’s components and the size of the space around the point.

The Laplace–Beltrami operator combines these ideas. It describes how a function spreads out over a curved space, using the divergence of the gradient.

Kulkarni–Nomizu product

The Kulkarni–Nomizu product is a way to combine two special mathematical objects, called symmetric covariant 2-tensors, to create a new one. This new object is a covariant 4-tensor. The formula shows how the values of the new tensor are calculated using the values of the original tensors. This product has a useful property: the order of the tensors does not change the result.

In an inertial frame

An orthonormal inertial frame is a special set of coordinates where, at the starting point, certain measurements become simpler. These coordinates are also called normal coordinates. In this special frame, some mathematical expressions for studying curved spaces become easier to understand, but they only work exactly at the starting point.

Conformal change

Let ( g ) be a Riemannian or pseudo-Riemannian metric on a smooth manifold ( M ), and ( \varphi ) a smooth real-valued function on ( M ). Then ( \tilde{g} = e^{2\varphi} g ) is also a Riemannian metric on ( M ). We say that ( \tilde{g} ) is (pointwise) conformal to ( g ).

Evidently, conformality of metrics is an equivalence relation. Here are some formulas for conformal changes in tensors associated with the metric. Quantities marked with a tilde will be associated with ( \tilde{g} ), while those unmarked with such will be associated with ( g ).

Levi-Civita connection

( \tilde{\Gamma}{ij}^k = \Gamma{ij}^k + \frac{\partial \varphi}{\partial x^i} \delta_j^k + \frac{\partial \varphi}{\partial x^j} \delta_i^k - \frac{\partial \varphi}{\partial x^l} g^{lk} g_{ij} )
( \tilde{\nabla}_X Y = \nabla_X Y + d\varphi(X) Y + d\varphi(Y) X - g(X, Y) \nabla \varphi )

(4,0) Riemann curvature tensor

( \tilde{R}{ijkl} = e^{2\varphi} R{ijkl} + e^{2\varphi} ( g_{ik} T_{jl} + g_{jl} T_{ik} - g_{il} T_{jk} - g_{jk} T_{il} ) ) where ( T_{ij} = \nabla_i \nabla_j \varphi - \nabla_i \varphi \nabla_j \varphi + \frac{1}{2} |d\varphi|^2 g_{ij} )

Using the Kulkarni–Nomizu product:

( \tilde{Rm} = e^{2\varphi} Rm + e^{2\varphi} g \wedge \bigcirc (\operatorname{Hess} \varphi - d\varphi \otimes d\varphi + \frac{1}{2} |d\varphi|^2 g ) )

Ricci tensor

( \tilde{R}{ij} = R{ij} - (n-2) ( \nabla_i \nabla_j \varphi - \nabla_i \varphi \nabla_j \varphi ) - ( \Delta \varphi + (n-2) |d\varphi|^2 ) g_{ij} )
( \tilde{Ric} = Ric - (n-2) ( \operatorname{Hess} \varphi - d\varphi \otimes d\varphi ) - ( \Delta \varphi + (n-2) |d\varphi|^2 ) g )

Scalar curvature

( \tilde{R} = e^{-2\varphi} R - 2(n-1) e^{-2\varphi} \Delta \varphi - (n-2)(n-1) e^{-2\varphi} |d\varphi|^2 )
if ( n \neq 2 ) this can be written ( \tilde{R} = e^{-2\varphi} [ R - \frac{4(n-1)}{(n-2)} e^{-(n-2)\varphi /2} \Delta ( e^{(n-2)\varphi /2} ) ] )

Traceless Ricci tensor

( \tilde{R}{ij} - \frac{1}{n} \tilde{R} \tilde{g}{ij} = R_{ij} - \frac{1}{n} R g_{ij} - (n-2) ( \nabla_i \nabla_j \varphi - \nabla_i \varphi \nabla_j \varphi ) + \frac{(n-2)}{n} ( \Delta \varphi - |d\varphi|^2 ) g_{ij} )
( \tilde{Ric} - \frac{1}{n} \tilde{R} \tilde{g} = Ric - \frac{1}{n} R g - (n-2) ( \operatorname{Hess} \varphi - d\varphi \otimes d\varphi ) + \frac{(n-2)}{n} ( \Delta \varphi - |d\varphi|^2 ) g )

(3,1) Weyl curvature

( \tilde{W}{ijk}^l = W{ijk}^l )
( \tilde{W}(X, Y, Z) = W(X, Y, Z) ) for any vector fields ( X, Y, Z )

Volume form

( \sqrt{\det \tilde{g}} = e^{n\varphi} \sqrt{\det g} )
( d\mu_{\tilde{g}} = e^{n\varphi} d\mu_g )

Hodge operator on ( p )-forms

( \tilde{\ast}{i_1 \cdots i{n-p}}^{j_1 \cdots j_p} = e^{(n-2p)\varphi} \ast_{i_1 \cdots i_{n-p}}^{j_1 \cdots j_p} )
( \tilde{\ast} = e^{(n-2p)\varphi} \ast )

Codifferential on ( p )-forms

( \tilde{d^{\ast}}{j_1 \cdots j{p-1}}^{i_1 \cdots i_p} = e^{-2\varphi} (d^{\ast}){j_1 \cdots j{p-1}}^{i_1 \cdots i_p} - (n-2p) e^{-2\varphi} \nabla^{i_1} \varphi \delta_{j_1}^{i_2} \cdots \delta_{j_{p-1}}^{i_p} )
( \tilde{d^{\ast}} = e^{-2\varphi} d^{\ast} - (n-2p) e^{-2\varphi} \iota_{\nabla \varphi} )

Laplacian on functions

( \tilde{\Delta} \Phi = e^{-2\varphi} ( \Delta \Phi + (n-2) g(d\varphi, d\Phi) ) )

Hodge Laplacian on ( p )-forms

( \tilde{\Delta^d} \omega = e^{-2\varphi} ( \Delta^d \omega - (n-2p) d \circ \iota_{\nabla \varphi} \omega - (n-2p-2) \iota_{\nabla \varphi} \circ d \omega + 2(n-2p) d\varphi \wedge \iota_{\nabla \varphi} \omega - 2 d\varphi \wedge d^{\ast} \omega ) )

The "geometer's" sign convention is used for the Hodge Laplacian here. In particular it has the opposite sign on functions as the usual Laplacian.

Second fundamental form of an immersion

Suppose ( (M, g) ) is Riemannian and ( F: \Sigma \to (M, g) ) is a twice-differentiable immersion. Recall that the second fundamental form is, for each ( p \in M ), a symmetric bilinear map ( h_p: T_p \Sigma \times T_p \Sigma \to T_{F(p)} M ), which is valued in the ( g_{F(p)} )-orthogonal linear subspace to ( dF_p(T_p \Sigma) \subset T_{F(p)} M ). Then

( \tilde{h}(u, v) = h(u, v) - (\nabla \varphi)^{\perp} g(u, v) ) for all ( u, v \in T_p M )

Here ( (\nabla \varphi)^{\perp} ) denotes the ( g_{F(p)} )-orthogonal projection of ( \nabla \varphi \in T_{F(p)} M ) onto the ( g_{F(p)} )-orthogonal linear subspace to ( dF_p(T_p \Sigma) \subset T_{F(p)} M ).

Mean curvature of an immersion

In the same setting as above (and suppose ( \Sigma ) has dimension ( n )), recall that the mean curvature vector is for each ( p \in \Sigma ) an element ( \mathbf{H}p \in T{F(p)} M ) defined as the ( g )-trace of the second fundamental form. Then

( \tilde{\mathbf{H}} = e^{-2\varphi} (\mathbf{H} - n (\nabla \varphi)^{\perp}) )

Note that this transformation formula is for the mean curvature vector, and the formula for the mean curvature ( H ) in the hypersurface case is

( \tilde{H} = e^{-\varphi} (H - n \langle \nabla \varphi, \eta \rangle ) ) where ( \eta ) is a (local) normal vector field.

Variation formulas

This section talks about formulas that describe how certain mathematical objects change when the shape of a space, called a manifold, is smoothly altered over time. These formulas are important in a branch of mathematics called Riemannian geometry, which studies the shapes and properties of curved spaces.

The formulas involve complex mathematical operations, but they help mathematicians understand how different measurements, like curvature and distance, change as the space deforms. These ideas are used in fields like physics, where they help describe the curvature of space-time in Einstein's theory of relativity.

Principal symbol

This section talks about special rules that help us understand how certain important math objects change in a special kind of space called a Riemannian manifold. These rules are called "principal symbols." They show us how things like the Riemann tensor, Ricci tensor, and scalar curvature behave when we look at very small details of the space.

The rules involve complicated math expressions, but they basically tell us how these objects respond to small changes in the space’s shape.