Complete manifold

In mathematics, a complete manifold (or geodesically complete manifold) M is a (pseudo-) Riemannian manifold for which, starting at any point p of M, there are straight paths extending infinitely in all directions.

Formally, a manifold

M

{\displaystyle M}

is (geodesically) complete if for any maximal geodesic

ℓ

:

I

→

M

{\displaystyle \ell :I\to M}

, it holds that

I

=

(

−

∞

,

∞

)

{\displaystyle I=(-\infty ,\infty )}

. A geodesic is maximal if its domain cannot be extended.

Equivalently,

M

{\displaystyle M}

is (geodesically) complete if for all points

p

∈

M

{\displaystyle p\in M}

, the exponential map at

p

{\displaystyle p}

is defined on

T

p

M

{\displaystyle T_{p}M}

, the entire tangent space at

p

{\displaystyle p}

.

Hopf–Rinow theorem

The Hopf–Rinow theorem gives alternative characterizations of completeness. Let

(

M

,

g

)

{\displaystyle (M,g)}

be a connected Riemannian manifold and let

d

g

:

M

×

M

→

[

0

,

∞

)

{\displaystyle d_{g}:M\times M\to [0,\infty )}

be its Riemannian distance function.

The Hopf–Rinow theorem states that

(

M

,

g

)

{\displaystyle (M,g)}

is (geodesically) complete if and only if it satisfies one of the following equivalent conditions:

The metric space

(

M

,

d

g

)

{\displaystyle (M,d_{g})}

is complete (every

d

g

{\displaystyle d_{g}}

-Cauchy sequence converges),

All closed and bounded subsets of

M

{\displaystyle M}

are compact.

Examples and non-examples

Euclidean space

R

n

{\displaystyle \mathbb {R} ^{n}}

, the sphere

S

n

{\displaystyle \mathbb {S} ^{n}}

, and the tori

T

n

{\displaystyle \mathbb {T} ^{n}}

(with their natural Riemannian metrics) are all complete manifolds.

All compact Riemannian manifolds and all homogeneous manifolds are geodesically complete. All symmetric spaces are geodesically complete.