In the math branches of

differential geometry Differential geometry is a mathematical discipline that studies the geometry of smooth shapes and smooth spaces, otherwise known as smooth manifolds. It uses the techniques of differential calculus, integral calculus, linear algebra and multili ...

and

vector calculus Vector calculus, or vector analysis, is concerned with differentiation and integration of vector fields, primarily in 3-dimensional Euclidean space \mathbb^3. The term "vector calculus" is sometimes used as a synonym for the broader subject ...

, the second

covariant derivative In mathematics, the covariant derivative is a way of specifying a derivative along tangent vectors of a manifold. Alternatively, the covariant derivative is a way of introducing and working with a connection on a manifold by means of a different ...

, 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 eleme ...

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 every po ...

''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 (i.e. affine connection) that preserves th ...

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

smooth Smooth may refer to: Mathematics * Smooth function, a function that is infinitely differentiable; used in calculus and topology * Smooth manifold, a differentiable manifold for which all the transition maps are smooth functions * Smooth algebrai ...

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 may ...

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 v ...

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 placeho ...

. 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 notion of torsion is a manner of characterizing a twist or screw of a moving frame around a curve. The torsion of a curve, as it appears in the Frenet–Serret formulas, for instance, quantifies the twist of a curve ...

is zero, so that

\nabla_uv-\nabla_vu

, we may use this fact to write

Riemann curvature tensor In the mathematical field of differential geometry, the Riemann curvature tensor or Riemann–Christoffel tensor (after Bernhard Riemann and Elwin Bruno Christoffel) is the most common way used to express the curvature of Riemannian manifolds. ...

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 The comma is a punctuation mark that appears in several variants in different languages. It has the same shape as an apostrophe or single closing quotation mark () in many typefaces, but it differs from them in being placed on the baseline (t ...

/math> we find :

(\nabla_u v - \nabla_v u)(f) =

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