In the math branches of

differential geometry Differential geometry is a Mathematics, mathematical discipline that studies the geometry of smooth shapes and smooth spaces, otherwise known as smooth manifolds. It uses the techniques of Calculus, single variable calculus, vector calculus, lin ...

and

vector calculus Vector calculus or vector analysis is a branch of mathematics concerned with the differentiation and integration of vector fields, primarily in three-dimensional Euclidean space, \mathbb^3. The term ''vector calculus'' is sometimes used as a ...

, the second

covariant derivative In mathematics and physics, covariance is a measure of how much two variables change together, and may refer to: Statistics * Covariance matrix, a matrix of covariances between a number of variables * Covariance or cross-covariance between ...

, or the second order covariant derivative, of a vector field is the derivative of its derivative with respect to another two

tangent vector In mathematics, a tangent vector is a vector that is tangent to a curve or surface at a given point. Tangent vectors are described in the differential geometry of curves in the context of curves in R''n''. More generally, tangent vectors are ...

fields.

Definition

Formally, given a (pseudo)-Riemannian

manifold In mathematics, a manifold is a topological space that locally resembles Euclidean space near each point. More precisely, an n-dimensional manifold, or ''n-manifold'' for short, is a topological space with the property that each point has a N ...

(''M'', ''g'') associated with a

vector bundle In mathematics, a vector bundle is a topological construction that makes precise the idea of a family of vector spaces parameterized by another space X (for example X could be a topological space, a manifold, or an algebraic variety): to eve ...

''E'' → ''M'', let ∇ denote the

Levi-Civita connection In Riemannian or pseudo-Riemannian geometry (in particular the Lorentzian geometry of general relativity), the Levi-Civita connection is the unique affine connection on the tangent bundle of a manifold that preserves the ( pseudo-) Riemannian ...

given by the metric ''g'', and denote by Γ(''E'') the space of the smooth

sections Section, Sectioning, or Sectioned may refer to: Arts, entertainment and media * Section (music), a complete, but not independent, musical idea * Section (typography), a subdivision, especially of a chapter, in books and documents ** Section sig ...

of the total space ''E''. Denote by ''T^*M'' the

cotangent bundle In mathematics, especially differential geometry, the cotangent bundle of a smooth manifold is the vector bundle of all the cotangent spaces at every point in the manifold. It may be described also as the dual bundle to the tangent bundle. This m ...

of ''M''. Then the second covariant derivative can be defined as the

composition Composition or Compositions may refer to: Arts and literature *Composition (dance), practice and teaching of choreography * Composition (language), in literature and rhetoric, producing a work in spoken tradition and written discourse, to include ...

of the two ∇s as follows: :

\Gamma(E) \stackrel \Gamma(T^*M \otimes E) \stackrel \Gamma(T^*M \otimes T^*M \otimes E).

For example, given vector fields ''u'', ''v'', ''w'', a second

can be written as :

(\nabla^2_ w)^a = u^c v^b \nabla_c \nabla_b w^a

by using

abstract index notation Abstract index notation (also referred to as slot-naming index notation) is a mathematical notation for tensors and spinors that uses indices to indicate their types, rather than their components in a particular basis. The indices are mere placeh ...

. It is also straightforward to verify that :

(\nabla_u \nabla_v w)^a = u^c \nabla_c v^b \nabla_b w^a = u^c v^b \nabla_c \nabla_b w^a + (u^c \nabla_c v^b) \nabla_b w^a = (\nabla^2_ w)^a + (\nabla_ w)^a.

Thus :

\nabla^2_ w = \nabla_u \nabla_v w - \nabla_ w.

When the

torsion tensor In differential geometry, the torsion tensor is a tensor that is associated to any affine connection. The torsion tensor is a bilinear map of two input vectors X,Y, that produces an output vector T(X,Y) representing the displacement within a t ...

is zero, so that

\nabla_uv-\nabla_vu

, we may use this fact to write

Riemann curvature tensor Georg Friedrich Bernhard Riemann (; ; 17September 182620July 1866) was a German mathematician who made profound contributions to mathematical analysis, analysis, number theory, and differential geometry. In the field of real analysis, he is mos ...

as :

R(u,v) w=\nabla^2_ w - \nabla^2_ w.

Similarly, one may also obtain the second covariant derivative of a function ''f'' as :

\nabla^2_ f = u^c v^b \nabla_c \nabla_b f = \nabla_u \nabla_v f - \nabla_ f.

Again, for the torsion-free Levi-Civita connection, and for any vector fields ''u'' and ''v'', when we feed the function ''f'' into both sides of :

\nabla_u v - \nabla_v u =, v /math>
we  find 
: (\nabla_u v - \nabla_v u)(f) =, v f) = u(v(f)) - v(u(f)). .
This can be rewritten as 
: \nabla_ f - \nabla_ f = \nabla_u \nabla_v f - \nabla_v \nabla_u f, so we have 
: \nabla^2_ f = \nabla^2_ f. That is, the value of the second covariant derivative of a function is independent on the order of taking derivatives.

Notes

Tensors in general relativity Riemannian geometry {{math-physics-stub