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.

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 help us describe how vectors change when we move on a curved surface. They are useful for understanding the shape of spaces that are not flat.

The Christoffel symbols are linked to the metric tensor, which measures distances on the surface. The covariant derivative is a way to calculate changes in vectors and tensors while considering the space's curvature. This helps us learn about how things move and how physical laws work on curved surfaces.

Curvature tensors

Curvature tensors are important tools in Riemannian geometry. This 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 curvature of a space by measuring how vectors change when moved along curves. Other important tensors come from it. These include the Ricci curvature, which simplifies the Riemann tensor. There is also scalar curvature, which reduces the Ricci tensor to a single number. This number 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. We find it by looking at tiny changes in the function’s value in every direction.

Divergence tells us how much a vector field spreads out from a point. We work it out using changes in the vector field’s parts and the size of the space around the point.

The Laplace–Beltrami operator brings these ideas together. It explains 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 math tools, called symmetric covariant 2-tensors, to make a new one. This new tool is a covariant 4-tensor. The formula shows how to find the values of the new tensor from the values of the old tensors. This product has a helpful feature: changing the order of the tensors does not change the result.

In an inertial frame

An orthonormal inertial frame is a special way to set up coordinates. These coordinates are also called normal coordinates. In this special frame, some math for studying curved spaces gets simpler, but it only works perfectly at the starting point.

Conformal change

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

Here are some formulas for conformal changes in tensors linked to the metric. Quantities marked with a tilde will be linked to ( \tilde{g} ), while those without will be linked to ( 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 show how some math ideas change when the shape of a space is smoothly changed over time. These formulas are important in Riemannian geometry, which studies the shapes and properties of curved spaces.

The formulas use complex math, but they help mathematicians learn how measurements like curvature and distance change as the space moves. These ideas are used in fields like physics to help explain the curvature of space-time in Einstein's theory of relativity.

Principal symbol

This section talks about special rules that help us understand how 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 closely at small details of the space.

The rules use complicated math expressions, but they tell us how these objects react to tiny changes in the space’s shape.