Levi-civita Connection

i.e., the torsion-free Connection on the Tangent Bundle preserving a given Riemannian Metric (or Pseudo-Riemannian Metric ).
The Fundamental Theorem Of Riemannian Geometry states that there is unique connection which satisfies these properties.

In the theory of Riemannian and Pseudo-Riemannian Manifold s the term Covariant Derivative is often used for the Levi-Civita connection. The components of this connection with respect to a system of local coordinates are called Christoffel Symbols .

FORMAL DEFINITION

Let

(M,g)

abla

is Levi-Civita connection if it satisfy the following conditions

#''Preserves metric'', i.e., for any vector fields

X

Y

Z

we have

Xg(Y,Z)=g(
abla_X Y,Z)+g(Y,
abla_X Z)

, where

Xg(Y,Z)

denotes the derivative of function

g(Y,Z)

along vector field

X

.
#'' Torsion -free'', i.e., for any vector fields

X

and

Y

we have

abla_XY-
abla_YX= where [X,Y

X

and

Y

.

DERIVATIVE ALONG CURVE

Levi-Civita connection defines also a derivative along curves, usually denoted by

D

.

Given a smooth Curve

\gamma

(M,g)

V

\gamma

its derivative is defined by
:

D_tV=
abla_{\dot\gamma(t)}V.

	CATEGORIES ABOUT LEVI-CIVITA CONNECTION

	riemannian geometry

	connection mathematics

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