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This is a glossary of some terms used in Riemannian geometry and metric geometry — it doesn't cover the terminology of differential topology.

The following articles may also be useful; they either contain specialised vocabulary or provide more detailed expositions of the definitions given below.

• Connection

• Curvature

• Metric space

• Riemannian manifold

See also:

• Glossary of general topology

• Glossary of differential geometry and topology

• List of differential geometry topics

Unless stated otherwise, letters X, Y, Z below denote metric spaces, M, N denote Riemannian manifolds, |xy| or | x y | X {\displaystyle |xy|_{X}} denotes the distance between points x and y in X. Italic word denotes a self-reference to this glossary.

A caveat: many terms in Riemannian and metric geometry, such as convex function, convex set and others, do not have exactly the same meaning as in general mathematical usage.

• A

• B

• C

• D

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• F

• G

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• I

• J

• K

• L

• M

• N

• O

• P

• Q

• R

• S

• T

• U

• V

• W

• References

Alexandrov space a generalization of Riemannian manifolds with upper, lower or integral curvature bounds (the last one works only in dimension 2).

Arc-wise isometry the same as path isometry.

Autoparallel the same as totally geodesic.cite-ref-1[1]

Barycenter, see center of mass.

Bi-Lipschitz map. A map f : X → → Y {\displaystyle f:X\to Y} is called bi-Lipschitz if there are positive constants c and C such that for any x and y in X

c | x y | X ≤ ≤ | f ( x ) f ( y ) | Y ≤ ≤ C | x y | X . {\displaystyle c|xy|_{X}\leq |f(x)f(y)|_{Y}\leq C|xy|_{X}.}

Boundary at infinity. In general, a construction that may be regarded as a space of directions at infinity. For geometric examples, see for instance hyperbolic boundary, Gromov boundary, visual boundary, Tits boundary, Thurston boundary. See also projective space and compactification.

Busemann function given a ray, γ : [0, ∞)→X, the Busemann function is defined by B γ γ ( p ) = lim t → → ∞ ∞ ( | γ γ ( t ) − − p | − − t ) . {\displaystyle B_{\gamma }(p)=\lim _{t\to \infty }(|\gamma (t)-p|-t).}

Cartan-Hadamard space is a complete, simply-connected, non-positively curved Riemannian manifold.

Cartan–Hadamard theorem is the statement that a connected, simply connected complete Riemannian manifold with non-positive sectional curvature is diffeomorphic to Rn via the exponential map; for metric spaces, the statement that a connected, simply connected complete geodesic metric space with non-positive curvature in the sense of Alexandrov is a (globally) CAT(0) space.

Cartan (Élie) The mathematician after whom Cartan-Hadamard manifolds, Cartan subalgebras, and Cartan connections are named (not to be confused with his son Henri Cartan).

Center of mass. A point q ∈ ∈ M {\textstyle q\in M} is called the center of masscite-ref-2[2] of the points p 1 , p 2 , … … , p k {\textstyle p_{1},p_{2},\dots ,p_{k}} if it is a point of global minimum of the function

f ( x ) = ∑ ∑ i | p i x | 2 . {\displaystyle f(x)=\sum _{i}|p_{i}x|^{2}.}

Such a point is unique if all distances | p i p j | {\displaystyle |p_{i}p_{j}|} are less than the convexity radius.

Complete manifold According to the Riemannian Hopf-Rinow theorem, a Riemannian manifold is complete as a metric space, if and only if all geodesics can be infinitely extended.

Conformal map is a map which preserves angles.

Conformally flat a manifold M is conformally flat if it is locally conformally equivalent to a Euclidean space, for example standard sphere is conformally flat.

Conjugate points two points p and q on a geodesic γ γ {\displaystyle \gamma } are called conjugate if there is a Jacobi field on γ γ {\displaystyle \gamma } which has a zero at p and q.

Convex function. A function f on a Riemannian manifold is a convex if for any geodesic γ γ {\displaystyle \gamma } the function f ∘ ∘ γ γ {\displaystyle f\circ \gamma } is convex. A function f is called λ λ {\displaystyle \lambda } -convex if for any geodesic γ γ {\displaystyle \gamma } with natural parameter t {\displaystyle t} , the function f ∘ ∘ γ γ ( t ) − − λ λ t 2 {\displaystyle f\circ \gamma (t)-\lambda t^{2}} is convex.

Convex A subset K of a Riemannian manifold M is called convex if for any two points in K there is a unique shortest path connecting them which lies entirely in K, see also totally convex.

Convexity radius at a point p {\textstyle p} of a Riemannian manifold is the supremum of radii of balls centered at p {\textstyle p} that are (totally) convex. The convexity radius of the manifold is the infimum of the convexity radii at its points; for a compact manifold this is a positive number.cite-ref-3[3] Sometimes the additional requirement is made that the distance function to p {\textstyle p} in these balls is convex.cite-ref-4[4]

Diameter of a metric space is the supremum of distances between pairs of points.

Developable surface is a surface isometric to the plane.

Dilation same as Lipschitz constant.

Exponential map Exponential map (Lie theory), Exponential map (Riemannian geometry)

Finsler metric A generalization of Riemannian manifolds where the scalar product on the tangent space is replaced by a norm.

First fundamental form for an embedding or immersion is the pullback of the metric tensor.

Geodesic is a curve which locally minimizes distance.

Geodesic equation is the differential equation whose local solutions are the geodesics.

Geodesic flow is a flow on a tangent bundle TM of a manifold M, generated by a vector field whose trajectories are of the form ( γ γ ( t ) , γ γ ′ ( t ) ) {\displaystyle (\gamma (t),\gamma '(t))} where γ γ {\displaystyle \gamma } is a geodesic.

Geodesic metric space is a metric space where any two points are the endpoints of a minimizing geodesic.

Hadamard space is a complete simply connected space with nonpositive curvature.

Holonomy group is the subgroup of isometries of the tangent space obtained as parallel transport along closed curves.

Horosphere a level set of Busemann function.

Hyperbolic geometry (see also Riemannian hyperbolic space)

Injectivity radius The injectivity radius at a point p of a Riemannian manifold is the supremum of radii for which the exponential map at p is a diffeomorphism. The injectivity radius of a Riemannian manifold is the infimum of the injectivity radii at all points.cite-ref-5[5] See also cut locus.

For complete manifolds, if the injectivity radius at p is a finite number r, then either there is a geodesic of length 2r which starts and ends at p or there is a point q conjugate to p (see conjugate point above) and on the distance r from p.cite-ref-6[6] For a closed Riemannian manifold the injectivity radius is either half the minimal length of a closed geodesic or the minimal distance between conjugate points on a geodesic.

Infranilmanifold Given a simply connected nilpotent Lie group N acting on itself by left multiplication and a finite group of automorphisms F of N one can define an action of the semidirect product N ⋊ ⋊ F {\displaystyle N\rtimes F} on N. An orbit space of N by a discrete subgroup of N ⋊ ⋊ F {\textstyle N\rtimes F} which acts freely on N is called an infranilmanifold. An infranilmanifold is finitely covered by a nilmanifold.cite-ref-7[7]

Isometric embedding is an embedding preserving the Riemannian metric.

Isometry is a surjective map which preserves distances.

Isoperimetric function of a metric space X {\textstyle X} measures "how efficiently rectifiable loops are coarsely contractible with respect to their length". For the Cayley 2-complex of a finite presentation, they are equivalent to the Dehn function of the group presentation. They are invariant under quasi-isometries.cite-ref-8[8]

Jacobi field A Jacobi field is a vector field on a geodesic γ which can be obtained on the following way: Take a smooth one parameter family of geodesics γ γ τ τ {\displaystyle \gamma _{\tau }} with γ γ 0 = γ γ {\displaystyle \gamma _{0}=\gamma } , then the Jacobi field is described by

J ( t ) = ∂ ∂ γ γ τ τ ( t ) ∂ ∂ τ τ | τ τ = 0 . {\displaystyle J(t)=\left.{\frac {\partial \gamma _{\tau }(t)}{\partial \tau }}\right|_{\tau =0}.}

Length metric the same as intrinsic metric.

Levi-Civita connection is a natural way to differentiate vector fields on Riemannian manifolds.

Lipschitz constant of a map is the infimum of numbers L such that the given map is L-Lipschitz.

Lipschitz convergence the convergence of metric spaces defined by Lipschitz distance.

Lipschitz distance between metric spaces is the infimum of numbers r such that there is a bijective bi-Lipschitz map between these spaces with constants exp(-r), exp(r).cite-ref-9[9]

Logarithmic map, or logarithm, is a right inverse of Exponential map.cite-ref-10[10]cite-ref-11[11]

Minimal surface is a submanifold with (vector of) mean curvature zero.

Mostow's rigidity In dimension ≥ ≥ 3 {\textstyle \geq 3} , compact hyperbolic manifolds are classified by their fundamental group.

Natural parametrization is the parametrization by length.cite-ref-12[12]

Net A subset S of a metric space X is called ϵ ϵ {\textstyle \epsilon } -net if for any point in X there is a point in S on the distance ≤ ≤ ϵ ϵ {\textstyle \leq \epsilon } .cite-ref-13[13] This is distinct from topological nets which generalize limits.

Nilmanifold: An element of the minimal set of manifolds which includes a point, and has the following property: any oriented S 1 {\displaystyle S^{1}} -bundle over a nilmanifold is a nilmanifold. It also can be defined as a factor of a connected nilpotent Lie group by a lattice.

Normal bundle: associated to an embedding of a manifold M into an ambient Euclidean space R N {\textstyle {\mathbb {R} }^{N}} , the normal bundle is a vector bundle whose fiber at each point p is the orthogonal complement (in R N {\textstyle {\mathbb {R} }^{N}} ) of the tangent space T p M {\textstyle T_{p}M} .

Nonexpanding map same as short map.

Orthonormal frame bundle is the bundle of bases of the tangent space that are orthonormal for the Riemannian metric.

Polyhedral space a simplicial complex with a metric such that each simplex with induced metric is isometric to a simplex in Euclidean space.

Principal curvature is the maximum and minimum normal curvatures at a point on a surface.

Principal direction is the direction of the principal curvatures.

Proper metric space is a metric space in which every closed ball is compact. Equivalently, if every closed bounded subset is compact. Every proper metric space is complete.cite-ref-14[14]

Quasi-convex subspace of a metric space X {\textstyle X} is a subset Y ⊆ ⊆ X {\textstyle Y\subseteq X} such that there exists K ≥ ≥ 0 {\textstyle K\geq 0} such that for all y , y ′ ∈ ∈ Y {\textstyle y,y'\in Y} , for all geodesic segment [ y , y ′ ] {\textstyle [y,y']} and for all z ∈ ∈ [ y , y ′ ] {\textstyle z\in [y,y']} , d ( z , Y ) ≤ ≤ K {\textstyle d(z,Y)\leq K} .cite-ref-15[15]

Quasigeodesic has two meanings; here we give the most common. A map f : I → → Y {\displaystyle f:I\to Y} (where I ⊆ ⊆ R {\displaystyle I\subseteq \mathbb {R} } is a subinterval) is called a quasigeodesic if there are constants K ≥ ≥ 1 {\displaystyle K\geq 1} and C ≥ ≥ 0 {\displaystyle C\geq 0} such that for every x , y ∈ ∈ I {\displaystyle x,y\in I}

1 K d ( x , y ) − − C ≤ ≤ d ( f ( x ) , f ( y ) ) ≤ ≤ K d ( x , y ) + C . {\displaystyle {1 \over K}d(x,y)-C\leq d(f(x),f(y))\leq Kd(x,y)+C.}

Note that a quasigeodesic is not necessarily a continuous curve.

Quasi-isometry. A map f : X → → Y {\displaystyle f:X\to Y} is called a quasi-isometry if there are constants K ≥ ≥ 1 {\displaystyle K\geq 1} and C ≥ ≥ 0 {\displaystyle C\geq 0} such that

1 K d ( x , y ) − − C ≤ ≤ d ( f ( x ) , f ( y ) ) ≤ ≤ K d ( x , y ) + C . {\displaystyle {1 \over K}d(x,y)-C\leq d(f(x),f(y))\leq Kd(x,y)+C.}

and every point in Y has distance at most C from some point of f(X). Note that a quasi-isometry is not assumed to be continuous. For example, any map between compact metric spaces is a quasi isometry. If there exists a quasi-isometry from X to Y, then X and Y are said to be quasi-isometric.

Radius of metric space is the infimum of radii of metric balls which contain the space completely.cite-ref-16[16]

Ray is a one side infinite geodesic which is minimizing on each interval.cite-ref-17[17]

Riemann The mathematician after whom Riemannian geometry is named.

Riemann curvature tensor is often defined as the (4, 0)-tensor of the tangent bundle of a Riemannian manifold ( M , g ) {\textstyle (M,g)} as R p ( X , Y , Z ) W = g p ( ∇ ∇ X ∇ ∇ Y Z − − ∇ ∇ Y ∇ ∇ X Z − − ∇ ∇ [ X , Y ] Z , W ) , {\displaystyle R_{p}(X,Y,Z)W={g_{p}({\nabla _{X}\nabla _{Y}Z-\nabla _{Y}\nabla _{X}Z-\nabla _{[X,Y]}Z,W})},} for p ∈ ∈ M {\textstyle p\in M} and X , Y , Z , W ∈ ∈ T p M {\textstyle X,Y,Z,W\in T_{p}M} (depending on conventions, X {\textstyle X} and Y {\textstyle Y} are sometimes switched).

Riemannian submanifold A differentiable sub-manifold whose Riemannian metric is the restriction of the ambient Riemannian metric (not to be confused with sub-Riemannian manifold).

Riemannian submersion is a map between Riemannian manifolds which is submersion and submetry at the same time.

Second fundamental form is a quadratic form on the tangent space of hypersurface, usually denoted by II, an equivalent way to describe the shape operator of a hypersurface,

II ( v , w ) = ⟨ ⟨ S ( v ) , w ⟩ ⟩ . {\displaystyle {\text{II}}(v,w)=\langle S(v),w\rangle .}

It can be also generalized to arbitrary codimension, in which case it is a quadratic form with values in the normal space.

Sectional curvature at a point p {\textstyle p} of a Riemannian manifold M {\textstyle M} along the 2-plane spanned by two linearly independent vectors u , v ∈ ∈ T p M {\textstyle u,v\in T_{p}M} is the number σ σ p ( V e c t ( u , v ) ) = R p ( u , v , v , u ) g p ( u , u ) g p ( v , v ) − − g p ( u , v ) 2 {\displaystyle \sigma _{p}({Vect}(u,v))={\frac {R_{p}(u,v,v,u)}{g_{p}(u,u)g_{p}(v,v)-g_{p}(u,v)^{2}}}} where R p {\textstyle R_{p}} is the curvature tensor written as R p ( X , Y , Z ) W = g p ( ∇ ∇ X ∇ ∇ Y Z − − ∇ ∇ Y ∇ ∇ X Z − − ∇ ∇ [ X , Y ] Z , W ) {\textstyle R_{p}(X,Y,Z)W={g_{p}({\nabla _{X}\nabla _{Y}Z-\nabla _{Y}\nabla _{X}Z-\nabla _{[X,Y]}Z,W})}} , and g p {\textstyle {g_{p}}} is the Riemannian metric.

Shape operator for a hypersurface M is a linear operator on tangent spaces, Sp: TpM→TpM. If n is a unit normal field to M and v is a tangent vector then

S ( v ) = ± ± ∇ ∇ v n {\displaystyle S(v)=\pm \nabla _{v}n}

(there is no standard agreement whether to use + or − in the definition).

Short map is a distance non increasing map.

Sol manifold is a factor of a connected solvable Lie group by a lattice.

Submetry A short map f between metric spaces is called a submetrycite-ref-18[18] if there exists R > 0 such that for any point x and radius r < R the image of metric r-ball is an r-ball, i.e. f ( B r ( x ) ) = B r ( f ( x ) ) . {\displaystyle f(B_{r}(x))=B_{r}(f(x)).} Sub-Riemannian manifold

Symmetric space Riemannian symmetric spaces are Riemannian manifolds in which the geodesic reflection at any point is an isometry. They turn out to be quotients of a real Lie group by a maximal compact subgroup whose Lie algebra is the fixed subalgebra of the involution obtained by differentiating the geodesic symmetry. This algebraic data is enough to provide a classification of the Riemannian symmetric spaces.

Systole The k-systole of M, s y s t k ( M ) {\textstyle syst_{k}(M)} , is the minimal volume of k-cycle nonhomologous to zero.

Thurston's geometries The eight 3-dimensional geometries predicted by Thurston's geometrization conjecture, proved by Perelman: S 3 {\textstyle \mathbb {S} ^{3}} , R × × S 2 {\textstyle \mathbb {R} \times \mathbb {S} ^{2}} , R 3 {\textstyle \mathbb {R} ^{3}} , R × × H 2 {\textstyle \mathbb {R} \times \mathbb {H} ^{2}} , H 3 {\textstyle \mathbb {H} ^{3}} , S o l {\displaystyle \mathrm {Sol} } , N i l {\displaystyle \mathrm {Nil} } , and P S L ~ ~ 2 ( R ) {\textstyle {\widetilde {PSL}}_{2}(\mathbb {R} )} .

Totally convex A subset K of a Riemannian manifold M is called totally convex if for any two points in K any geodesic connecting them lies entirely in K, see also convex.cite-ref-19[19]

Totally geodesic submanifold is a submanifold such that all geodesics in the submanifold are also geodesics of the surrounding manifold.cite-ref-20[20]

Uniquely geodesic metric space is a metric space where any two points are the endpoints of a unique minimizing geodesic.

Word metric on a group is a metric of the Cayley graph constructed using a set of generators.

cite-note-11. ↑ citerefkobayashinomizu1963Kobayashi, Shōshichi; Nomizu, Katsumi (1963). "Chapter VII Submanifolds, 8. Autoparallel submanifolds and totally geodesic submanifolds". Foundations of differential geometry. Interscience Publishers, New York, NY. pp. 53–62. ISBN 978-0-471-15732-8. Zbl 0175.48504. {{cite book}}: ISBN / Date incompatibility (help)

cite-note-22. ↑ citerefmancinellipuppo2023Mancinelli, Claudio; Puppo, Enrico (2023-06-01). "Computing the Riemannian center of mass on meshes". Computer Aided Geometric Design. 103 102203. doi:10.1016/j.cagd.2023.102203. ISSN 0167-8396.

cite-note-33. ↑ citerefgallothulinlafontaine2004Gallot, Sylvestre; Hulin, Dominique; Lafontaine, Jacques (2004), Gallot, Sylvestre; Hulin, Dominique; Lafontaine, Jacques (eds.), "Riemannian metrics", Riemannian Geometry, Berlin, Heidelberg: Springer, Remark after Proof of Corollary 2.89, p.87, doi:10.1007/978-3-642-18855-8_2, ISBN 978-3-642-18855-8, retrieved 2024-11-28

cite-note-44. ↑ citerefpetersen2016Petersen, Peter (2016), Petersen, Peter (ed.), "Sectional Curvature Comparison I", Riemannian Geometry, Graduate Texts in Mathematics, vol. 171, Cham: Springer International Publishing, Theorem 6.4.8, pp. 258-259, doi:10.1007/978-3-319-26654-1_6, ISBN 978-3-319-26654-1, retrieved 2024-11-29

cite-note-55. ↑ citereflee2009Lee, Jeffrey M. (2009). "13. Riemannian and Semi-Riemannian Geometry, Definition 13.141". Manifolds and differential geometry. Providence, RI: American Mathematical Society (AMS). p. 615. ISBN 978-0-8218-4815-9. Zbl 1190.58001.

cite-note-66. ↑ citerefgallothulinlafontaine2004Gallot, Sylvestre; Hulin, Dominique; Lafontaine, Jacques (2004), Gallot, Sylvestre; Hulin, Dominique; Lafontaine, Jacques (eds.), "Curvature", Riemannian Geometry, Berlin, Heidelberg: Springer, Scholium 3.78, doi:10.1007/978-3-642-18855-8_3, ISBN 978-3-642-18855-8, retrieved 2024-11-28

cite-note-77. ↑ citerefhirsch1970Hirsch, Morris W. (1970). "Expanding maps and transformation groups". Global Analysis. Proceedings of Symposia in Pure Mathematics. Vol. 14. pp. 125–131. doi:10.1090/pspum/014/0298701. ISBN 978-0-8218-1414-7. Zbl 0223.58009.

cite-note-88. ↑ citerefbridsonhaefliger1999Bridson, Martin R.; Haefliger, André (1999), Bridson, Martin R.; Haefliger, André (eds.), "δ-Hyperbolic Spaces and Area", Metric Spaces of Non-Positive Curvature, Berlin, Heidelberg: Springer, 2. Area and isoperimetric inequalities, pp. 414 – 417, doi:10.1007/978-3-662-12494-9_21, ISBN 978-3-662-12494-9, retrieved 2024-12-23

cite-note-99. ↑ citerefburagoburagoivanov2001Burago, Dmitri; Burago, Yurii; Ivanov, Sergei (2001). A course in metric geometry. Providence, RI: American Mathematical Society (AMS). Chapter 7, §7.2, pp. 249-250. ISBN 0-8218-2129-6. Zbl 0981.51016.

cite-note-1010. ↑ citerefburagoburagoivanov2001Burago, Dmitri; Burago, Yurii; Ivanov, Sergei (2001). A course in metric geometry. Providence, RI: American Mathematical Society (AMS). Chapter 9, §9.1, pp. 321-322. ISBN 0-8218-2129-6. Zbl 0981.51016.

cite-note-1111. ↑ citereflang1999Lang, Serge (1999). "Fundamentals of Differential Geometry". Graduate Texts in Mathematics. 191. Chapter XII An example of seminegative curvature, p. 323. doi:10.1007/978-1-4612-0541-8. ISBN 978-1-4612-6810-9. ISSN 0072-5285.

cite-note-1212. ↑ citerefburagoburagoivanov2001Burago, Dmitri; Burago, Yurii; Ivanov, Sergei (2001). A course in metric geometry. Providence, RI: American Mathematical Society (AMS). Chapter 2, §2.5.1, Definition 2.5.7. ISBN 0-8218-2129-6. Zbl 0981.51016.

cite-note-1313. ↑ citerefburagoburagoivanov2001Burago, Dmitri; Burago, Yurii; Ivanov, Sergei (2001). A course in metric geometry. Providence, RI: American Mathematical Society (AMS). Chapter 1, §1.6, Definition 1.6.1, p. 13. ISBN 0-8218-2129-6. Zbl 0981.51016.

cite-note-1414. ↑ citerefbridsonhaefliger1999Bridson, Martin R.; Haefliger, André (1999), Bridson, Martin R.; Haefliger, André (eds.), "Basic Concepts", Metric Spaces of Non-Positive Curvature, Berlin, Heidelberg: Springer, Chapter I.1, § Metric spaces, Definitions 1.1, p. 2, doi:10.1007/978-3-662-12494-9_1, ISBN 978-3-662-12494-9, retrieved 2024-11-29

cite-note-1515. ↑ citerefbridsonhaefliger1999Bridson, Martin R.; Haefliger, André (1999), Bridson, Martin R.; Haefliger, André (eds.), "Non-Positive Curvature and Group Theory", Metric Spaces of Non-Positive Curvature, Berlin, Heidelberg: Springer, Definition 3.4, p. 460, doi:10.1007/978-3-662-12494-9_22, ISBN 978-3-662-12494-9, retrieved 2024-12-23

cite-note-1616. ↑ citerefburagoburagoivanov2001Burago, Dmitri; Burago, Yurii; Ivanov, Sergei (2001). A course in metric geometry. Providence, RI: American Mathematical Society (AMS). Chapter 10, §10.4, Exercise 10.4.5, p. 366. ISBN 0-8218-2129-6. Zbl 0981.51016.

cite-note-1717. ↑ citerefpetersen2016Petersen, Peter (2016). "Riemannian Geometry". Graduate Texts in Mathematics. 171. Chapter 7, §7.3.1 Rays and Lines, p. 298. doi:10.1007/978-3-319-26654-1. ISBN 978-3-319-26652-7. ISSN 0072-5285.

cite-note-1818. ↑ citerefberestovskii1987Berestovskii, V. N. (1987-07-01). "Submetries of space-forms of negative curvature". Siberian Mathematical Journal. 28 (4): 552–562. Bibcode:1987SibMJ..28..552B. doi:10.1007/BF00973842. ISSN 1573-9260.

cite-note-1919. ↑ citerefpetersen2016Petersen, Peter (2016). "Riemannian Geometry". Graduate Texts in Mathematics. 171. Chapter 12, §12.4 The Soul Theorem, p. 463. doi:10.1007/978-3-319-26654-1. ISBN 978-3-319-26652-7. ISSN 0072-5285.

cite-note-2020. ↑ citerefgallothulinlafontaine2004Gallot, Sylvestre; Hulin, Dominique; Lafontaine, Jacques (2004). "Riemannian Geometry". Universitext. Chapter 2, §2.C.1, Definition 2.80 bis, p.82. doi:10.1007/978-3-642-18855-8. ISBN 978-3-540-20493-0. ISSN 0172-5939.